Sample Writeup

Evaluation of BMI Prediction Models Among Patients Who Have Undergone a Stroke Evaluation

Evaluation of BMI Prediction Models Among Patients Who Have Undergone a Stroke Evaluation

Preliminary information

Because patients who are monitored for stroke status during hospital visits are often watched closely and put through a myriad of tests, they may have exceptionally rich EHR data and be useful clinical research subjects. The purpose of this study is to use EHR data pulled from patients who had been evaluated for stroke to create a useful model to predict BMI.

1 Setup and Data Ingest

1.1 Setup

The packages and parameters used in this code chunk are present in order to generate a legible report.

library(knitr)
library(rmdformats)
library(rmarkdown)

options(max.print="100")
knitr::opts_chunk$set(echo = TRUE)
knitr::opts_chunk$set(comment = NA)
options(width = 70)

1.2 Loading Necessary R Packages

Each of the following packages were necessary for the analysis and/or graphical display of information used in this project.

setwd(dirname(getwd()))
source("Love-boost.R")
library(janitor)
library(Epi)
library(kableExtra)
library(GGally)
library(ggforce)
library(ggstance)
library(modelsummary)
library(ggdist)
library(gghalves)
library(ggmosaic)
library(car)
library(tidyr)
library(magrittr)
library(mosaic)
library(equatiomatic)
library(simputation) 
library(patchwork)
library(broom)
library(naniar)
library(tidyverse)



theme_set(theme_bw())

1.3 Data ingest

The following code is present for the purpose of ingesting the raw data, which is a .csv file. Additionally, data types were standardized in this step by converting necessary character variables into factors and retaining data for bmi as numeric. The id variable is maintained as a character variable here as it is just a unique numeric identifier for each patient. It will only be used for identification purposes in subsequent analyses, not as a quantitative or categorical variable that affects the outcome of interest.

stroke_raw <- read.csv("STROKEDATA.csv") |> 
   mutate(across(where(is.character), as.factor)) |>
  mutate(id = as.character(id))  |> 
  mutate(across(where(is.integer), as.factor)) |> 
  mutate(bmi = as.character(bmi)) |> 
  mutate(bmi = as.numeric(bmi))

2 Cleaning the Data

The following code is present for the purpose of creating a preliminary data frame in the form of a tibble that contains complete cases of each of the variables being analyzed in this project. It should be noted that only employed individuals are being considered in this study, and for that reason, two categories are omitted from the work type variable. This code also renames the components of the stroke variable and re-levels the components of the smoking_status variable in order to have more communicable results in subsequent analyses. Finally, a brief summary of the analytic tibble with complete cases, named stroke_complete_cases is provided.

stroke_raw <- stroke_raw |> 
  filter(complete.cases(work_type)) |> 
  filter(work_type == "Govt_job" | 
           work_type == "Private" | 
           work_type == "Self-employed") |> 
  droplevels()


stroke_raw$smoking_status <- recode_factor(stroke_raw$smoking_status, 
                                             "Unknown" = "N/A")

stroke_complete_cases <- stroke_raw |>
 filter(complete.cases(id, 
                       age, 
                       work_type, 
                       Residence_type, 
                       avg_glucose_level, 
                       bmi, 
                       smoking_status, stroke),  
        bmi != "N/A", smoking_status != "N/A", gender != "Other") 
    

stroke_complete_cases$stroke <- recode_factor(stroke_complete_cases$stroke, 
                                       "0" = "No Stroke",
                                             "1" = "Stroke")


stroke_complete_cases <- stroke_complete_cases |> 
  mutate(smoking_status = fct_relevel(smoking_status, 
                                      "never smoked",
                                      "formerly smoked", 
                                      "smokes")) |> 
  droplevels()


stroke_complete_cases <- stroke_complete_cases |> 
  as_tibble(stroke_complete_cases)



glimpse(stroke_complete_cases)
Rows: 3,343
Columns: 12
$ id                <chr> "9046", "31112", "60182", "1665", "56669",…
$ gender            <fct> Male, Male, Female, Female, Male, Male, Fe…
$ age               <dbl> 67, 80, 49, 79, 81, 74, 69, 81, 61, 54, 79…
$ hypertension      <fct> 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, …
$ heart_disease     <fct> 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, …
$ ever_married      <fct> Yes, Yes, Yes, Yes, Yes, Yes, No, Yes, Yes…
$ work_type         <fct> Private, Private, Private, Self-employed, …
$ Residence_type    <fct> Urban, Rural, Urban, Rural, Urban, Rural, …
$ avg_glucose_level <dbl> 228.69, 105.92, 171.23, 174.12, 186.21, 70…
$ bmi               <dbl> 36.6, 32.5, 34.4, 24.0, 29.0, 27.4, 22.8, …
$ smoking_status    <fct> formerly smoked, never smoked, smokes, nev…
$ stroke            <fct> Stroke, Stroke, Stroke, Stroke, Stroke, St…

Checking to make sure that none of the values for bmi are unrealistically high or low:

The following code is present for the purpose of displaying the number of bmi observations above 50 (10 points above the clinical level of morbid obesity as per CDC guidance) and below 10 (8 points below the clinical underweight cutoff point as per the CDC). This allows for any abnormally large or small values in the data set to be noted so that they can be omitted.

Checking the high-end of the BMI variable::

sum(stroke_complete_cases$bmi > 50) 
[1] 58

Checking the low-end of the BMI variable:

sum(stroke_complete_cases$bmi < 10)
[1] 0

Although the lowest bmi value in the stroke_complete_cases data frame appears to be in a reasonable area for a low BMI per CDC guidance, there are a total of 58 recorded BMI values over 50, which is considerably above the widely accepted cutoff value for morbid obesity of 40 (as per CDC guidance). For this reason values of bmi over 50 will be removed from the data frame, as they are either serious anomalies or possibly improperly measured/reported values. The following code removes these values and rechecks the number of abnormal values:

stroke_complete_cases <- stroke_complete_cases[which(stroke_complete_cases$bmi < 50),]

sum(stroke_complete_cases$bmi > 50) 
[1] 0
sum(stroke_complete_cases$bmi < 10)
[1] 0

2.1 Checking the key outcome (BMI) and key predictor (Average Blood Glucose Level)

The following code is present for the purpose of displaying a numeric summary for the key outcome of the study (bmi) and the key predictor of the study (avg_glucose_level).

df_stats(~bmi + avg_glucose_level, data = stroke_complete_cases) |> 
  kbl(caption = 
        "Numerical Summary of Key Outcome 
      (BMI) and Key Predictor (Average Blood Glucose Level)", digits = 2) |> 
  kable_styling(font_size = 20) |> 
  kable_paper("hover", full_width = F)
Numerical Summary of Key Outcome (BMI) and Key Predictor (Average Blood Glucose Level)
response min Q1 median Q3 max mean sd n missing
bmi 11.50 25.30 29.10 33.90 49.90 30.00 6.38 3285 0
avg_glucose_level 55.12 77.19 92.27 115.98 271.74 108.23 47.73 3285 0

As expected, there are no missing values for either of the variables above. Additionally, it can be seen that all values for BMI are between 10 and 50 and the interquartile range of average blood glucose level is below 140, demonstrating that neither of these variables contain abnormal values by the clinical standards reported by the CDC.

2.2 Checking the Quantitative Predictors:

For this analysis, there is only one quantitative predictor that is not the key predictor (avg_glucose_level), which is age. The following code provides a numeric summary of age.

df_stats(~age, data = stroke_complete_cases) |> 
  kbl(caption = "Numerical summary of quantitative predictor (Age)", 
      digits = 2) |> kable_styling(font_size = 20) |> 
  kable_paper("hover", 
              full_width = F)
Numerical summary of quantitative predictor (Age)
response min Q1 median Q3 max mean sd n missing
age 13 35 50 64 82 49.59 18.29 3285 0

It can be seen above that a wide range of ages will be included in this study, including ages as low as 13 and as high as 82. The average and median ages appear to be around 50.

2.3 Checking the Categorical Variables:

In this section, numeric summaries each of the categorical variables are individually provided. Each of these variables are to be used as a predictor for the key outcome, bmi.

2.3.1 Residence type

The following code provides a numeric summary of the Residence_type variable:

stroke_complete_cases |> 
  tabyl(Residence_type) |> 
  kbl(caption = "Summary of Residence Type", digits = 2) |> 
  kable_styling(font_size = 20) |> 
  kable_paper("hover", full_width = F)
Summary of Residence Type
Residence_type n percent
Rural 1613 0.49
Urban 1672 0.51

It can be seen in the above table that there is nearly a 50/50 split between people included in this study that live in a rural environment and people included in this study that live in an urban environment.

2.3.2 Stroke

The following code provides a numeric summary of the stroke variable:

stroke_complete_cases |> tabyl(stroke) |> 
  kbl(caption = "Summary of Stroke Status",digits = 2) |> 
  kable_styling(font_size = 20) |> 
  kable_paper("hover", 
              full_width = F)
Summary of Stroke Status
stroke n percent
No Stroke 3106 0.95
Stroke 179 0.05

The above table demonstrates that only about five percent of the people included in this study had suffered a stroke in their lifetime. It should be noted that everybody in this study was evaluated for a stroke, however.

2.3.3 Work Type

The following code provides a numeric summary of the work_type variable:

stroke_complete_cases |> 
  tabyl(work_type) |> 
  kbl(caption = "Summary of Employment Category",digits = 2) |> 
  kable_styling(font_size = 20) |> 
  kable_paper("hover", full_width = F)
Summary of Employment Category
work_type n percent
Govt_job 505 0.15
Private 2159 0.66
Self-employed 621 0.19

The above table demonstrates that most people included in this study are privately employed. Around 15% of participants appear to be self-employed and 19% appear to be government employees.

2.3.4 Smoking Status

The following code provides a numeric summary of the smoking_status variable:

stroke_complete_cases |> 
  tabyl(smoking_status) |> 
  kbl(caption = "Summary of Smoking Status", digits = 2) |> 
  kable_styling(font_size = 20) |> 
  kable_paper("hover", full_width = F)
Summary of Smoking Status
smoking_status n percent
never smoked 1751 0.53
formerly smoked 811 0.25
smokes 723 0.22

It is demonstrated in the table above that close to half of the participants in this study have never smoked in their life, while the remaining portion is split fairly evenly between former smokers and current smokers. It appears that there are more former smokers than current smokers.

2.4 Evaluation of missing cases

The following code is present for the purpose of consolidating information on all missing values in one place, such that it can be determined that the stroke_complete_cases tibble can be considered to be made up of only complete cases. It should be noted that any missing cases are considered to have been missing completely at random (MCAR).

miss_var_summary(stroke_complete_cases)
# A tibble: 12 × 3
   variable          n_miss pct_miss
   <chr>              <int>    <dbl>
 1 id                     0        0
 2 gender                 0        0
 3 age                    0        0
 4 hypertension           0        0
 5 heart_disease          0        0
 6 ever_married           0        0
 7 work_type              0        0
 8 Residence_type         0        0
 9 avg_glucose_level      0        0
10 bmi                    0        0
11 smoking_status         0        0
12 stroke                 0        0

2.5 Printing The analytic tibble

The following code is present for the purpose of renaming the data set containing all of the six variables and printing it to show the final analytic tibble that will be used in all subsequent analyses.

Stroke_Analysis_Data <- stroke_complete_cases |> 
  select(id, age, bmi, 
         avg_glucose_level, work_type, Residence_type, 
         smoking_status, stroke)

Stroke_Analysis_Data
# A tibble: 3,285 × 8
   id      age   bmi avg_glucose_level work_t…¹ Resid…² smoki…³ stroke
   <chr> <dbl> <dbl>             <dbl> <fct>    <fct>   <fct>   <fct> 
 1 9046     67  36.6             229.  Private  Urban   former… Stroke
 2 31112    80  32.5             106.  Private  Rural   never … Stroke
 3 60182    49  34.4             171.  Private  Urban   smokes  Stroke
 4 1665     79  24               174.  Self-em… Rural   never … Stroke
 5 56669    81  29               186.  Private  Urban   former… Stroke
 6 53882    74  27.4              70.1 Private  Rural   never … Stroke
 7 10434    69  22.8              94.4 Private  Urban   never … Stroke
 8 12109    81  29.7              80.4 Private  Rural   never … Stroke
 9 12095    61  36.8             120.  Govt_job Rural   smokes  Stroke
10 12175    54  27.3             105.  Private  Urban   smokes  Stroke
# … with 3,275 more rows, and abbreviated variable names ¹​work_type,
#   ²​Residence_type, ³​smoking_status

3 Codebook and Data Description

3.1 Description of the subjects of this study

The data used for this study was taken from the electronic health records (EHRs) of 3285 persons aged 10-82 who, within their EHR, had explicitly been noted as either having suffered a stroke in their life or having not suffered a stroke in their life. All EHR data used in this study was de-identified and made public by McKinsey and Company, a consulting firm with oversight over a number of US hospital EHRs. All subjects had complete data for each of the variables listed below.

3.2 Codebook

Description of Variables

The following table contains the name of each variable in the Stroke_Analysis_Data data set used for this analysis, the type of variable it is, and a brief description of the levels for each variable. The variable types are denoted as such:

ID = Identification number

Quant = Quantitative variables

Binary = Two-category variables

X-cat = Multi-category variables) where X indicates the number of levels in the variable.

Variable Type Description / Levels
id ID A unique identification number assigned to each participant
bmi Quant Outcome Variable: Current body mass index of the patient as indicated by that patient’s EHR. Body mass index is calculated by dividing weight in pounds (lb) by height in inches (in) squared and multiplying that value by a conversion factor of 703.
work_type 3-Cat The category of work of the patient as indicated by that patient’s EHR. “Govt_job” indicates that the patient works for the government (public sector), “Private” indicates that the patient is employed and works for a private company (private sector), and “Self-employed” indicates that the patient is self employed.
stroke Binary Whether or not electronic medical records indicated that the individual had suffered a stroke at some point in their life. “Stroke” indicates that they suffered a stroke, “No Stroke” indicates that they did not.
smoking_status 3-Cat The smoking status of the patient as indicated by that patient’s EHR. “never smoked” indicates that the patient has no history of smoking, “formerly smoked” indicates that they have, but no longer smoke, and “smokes” indicates that the patient currently smokes.
Residence_type 2-cat The residence type in which the patient lives. Urban refers to an urban environment and Rural refers to a rural environment.
age Quant The age of the patient in years as indicated in that patient’s EHR.
avg_glucose_level Quant Key Predictor The average blood glucose level for the patient, measured in milligrams per deciliter (Mg/Dl) as indicated in that patient’s EHR

3.3 Analytic Tibble

The following code is present for the purpose of printing the analytic tibble in order to show that it is, in fact, a tibble.

Stroke_Analysis_Data
# A tibble: 3,285 × 8
   id      age   bmi avg_glucose_level work_t…¹ Resid…² smoki…³ stroke
   <chr> <dbl> <dbl>             <dbl> <fct>    <fct>   <fct>   <fct> 
 1 9046     67  36.6             229.  Private  Urban   former… Stroke
 2 31112    80  32.5             106.  Private  Rural   never … Stroke
 3 60182    49  34.4             171.  Private  Urban   smokes  Stroke
 4 1665     79  24               174.  Self-em… Rural   never … Stroke
 5 56669    81  29               186.  Private  Urban   former… Stroke
 6 53882    74  27.4              70.1 Private  Rural   never … Stroke
 7 10434    69  22.8              94.4 Private  Urban   never … Stroke
 8 12109    81  29.7              80.4 Private  Rural   never … Stroke
 9 12095    61  36.8             120.  Govt_job Rural   smokes  Stroke
10 12175    54  27.3             105.  Private  Urban   smokes  Stroke
# … with 3,275 more rows, and abbreviated variable names ¹​work_type,
#   ²​Residence_type, ³​smoking_status

3.4 Numerical Data Description:

The following code is present for the purpose of displaying a brief numeric description of each of the variables used in the Stroke_Analysis_Data data set.

Hmisc::describe(Stroke_Analysis_Data) 
Stroke_Analysis_Data 

 8  Variables      3285  Observations
----------------------------------------------------------------------
id 
       n  missing distinct 
    3285        0     3285 

lowest : 10056 10119 10133 10138 10139, highest: 9730  9752  9923  9926  9955 
----------------------------------------------------------------------
age 
       n  missing distinct     Info     Mean      Gmd      .05 
    3285        0       70        1    49.59    21.08       20 
     .10      .25      .50      .75      .90      .95 
      24       35       50       64       76       79 

lowest : 13 14 15 16 17, highest: 78 79 80 81 82
----------------------------------------------------------------------
bmi 
       n  missing distinct     Info     Mean      Gmd      .05 
    3285        0      318        1       30    7.144    21.00 
     .10      .25      .50      .75      .90      .95 
   22.40    25.30    29.10    33.90    39.06    42.20 

lowest : 11.5 14.1 15.0 15.7 16.0, highest: 49.3 49.4 49.5 49.8 49.9
----------------------------------------------------------------------
avg_glucose_level 
       n  missing distinct     Info     Mean      Gmd      .05 
    3285        0     2806        1    108.2    48.15    60.69 
     .10      .25      .50      .75      .90      .95 
   65.85    77.19    92.27   115.98   199.32   220.19 

lowest :  55.12  55.22  55.25  55.27  55.32
highest: 266.59 267.60 267.61 267.76 271.74
----------------------------------------------------------------------
work_type 
       n  missing distinct 
    3285        0        3 
                                                    
Value           Govt_job       Private Self-employed
Frequency            505          2159           621
Proportion         0.154         0.657         0.189
----------------------------------------------------------------------
Residence_type 
       n  missing distinct 
    3285        0        2 
                      
Value      Rural Urban
Frequency   1613  1672
Proportion 0.491 0.509
----------------------------------------------------------------------
smoking_status 
       n  missing distinct 
    3285        0        3 
                                                          
Value         never smoked formerly smoked          smokes
Frequency             1751             811             723
Proportion           0.533           0.247           0.220
----------------------------------------------------------------------
stroke 
       n  missing distinct 
    3285        0        2 
                              
Value      No Stroke    Stroke
Frequency       3106       179
Proportion     0.946     0.054
----------------------------------------------------------------------

4 My Research Question

Body mass index is a widely utilized reporting method for the relationship between the height and weight of a person. Values for BMI that are 20 and over 25 have been associated with higher all-cause mortality, with increasing risk as the distance from the 20–25 range increases. Given the positive correlation between BMI and all-cause mortality risk, models that are able to accurately predict BMI may be useful for determining at-risk patients who may need additional care and resources in the future. Average blood glucose level is an easy to measure metric that can be determined at nearly any clinic, thus it may be advantageous to see if it can be used to predict BMI value.

My research question is as follows:

How effectively can the body mass index of a hospital patient who has been evaluated for a stroke be predicted by the average blood glucose level of the patient that was recorded in their EHR and how is the quality of prediction changed when adjusting for age, employment type, residence type, stroke history, and smoking status?

5 Partitioning the Data

The following code is present for the purpose of randomly segregating the data from Stroke_Analysis_Data into a “training” data frame containing 70% of the observations called Stroke_Analysis_Train, and a second “test” data frame containing the remaining 30% of the observations, called Stroke_Analysis_Test.

set.seed(538) 

Stroke_Analysis_Train <- Stroke_Analysis_Data |> 
  slice_sample(prop = .70)
Stroke_Analysis_Test <- anti_join(Stroke_Analysis_Data, 
                                  Stroke_Analysis_Train, 
                                  by = "id")

dim(Stroke_Analysis_Data)
[1] 3285    8
dim(Stroke_Analysis_Train)
[1] 2299    8
dim(Stroke_Analysis_Test)
[1] 986   8

It can be seen above by the dimensions of the Stroke_Analysis_Data, Stroke_Analysis_Train, and Stroke_Analysis_Test data frames respectively, that each data frame has eight variables, and that 70% of the number of observations in Stroke_Analysis_Data are included in the Stroke_Analysis_Train data frame. 30% of the number of observations from Stroke_Analysis_Data are included in the Stroke_Analysis_Test data frame.

6 Transforming The Outcome

In this section, the properties of the training data set (Stroke_Analysis_Train) will be assessed for the purpose of determining whether or not a transformation is necessary before creating a linear regression model using the data.

6.1 Visualization of Stroke_Analysis_Train with no applied transformations

The following code creates a histogram of the distribution of BMI values across the Stroke_Analysis_Train data.

ggplot(Stroke_Analysis_Train, aes(x = bmi)) +
    geom_histogram(bins = 75, fill = "lightsalmon", color = "coral2") +
  guides("none") +
    labs(title = "Histogram of BMI Values by Employment Category",
         x = "BMI") + theme(
  panel.background = element_rect(fill = "lemonchiffon1",
                                colour = "lemonchiffon1",
                                size = 0.5, linetype = "solid")) + 
  theme(plot.background = element_rect(fill = "azure2"))

It can be seen in the above histogram that the distribution of BMI within the Stroke_Analysis_Train data set appears to be fairly normal with some right skew.

The following code is present for the purpose of creating a box and violin plot juxtaposed with normal QQ plots for bmi, allowing for further visualization of the distribution of values in the Stroke_Analysis_Train data set.

p1 <- ggplot(Stroke_Analysis_Train, aes(x ="",  
                                        y = bmi)) + 
  geom_violin(col = "coral2", 
              fill = "lightsalmon", 
              alpha = 0.3) +
  geom_boxplot(width = 0.3, notch = TRUE, 
               color = "coral2", fill = "azure2", ) +
  guides(fill = "none") +
  labs(title = "BMI Values",
       subtitle = "For the `Stroke_Analysis_Train` data set.",
       x = "`Stroke_Analysis_Train` data set", 
       y = "Body Mass Index") + 
  theme(plot.background = element_rect(fill = "azure2"))

p2 <- ggplot(Stroke_Analysis_Train, 
             aes(sample = bmi, 
                 col = "coral2")) +
  geom_qq(fill = "lightsalmon") +
  geom_qq_line(col = "darkturquoise") +
  guides(col = "none") +
  theme_bw() +
  labs(y = "BMI ",
       title = "Normal QQ Plot", 
       subtitle = "For the `Stroke_Analysis_Train` data set.") +
  theme(plot.background = element_rect(fill = "azure2"))


(p1 + theme(
  panel.background = element_rect(fill = "lemonchiffon1",
                                colour = "lemonchiffon1",
                                size = 0.5, linetype = "solid"))) + (p2  +
    plot_annotation(title = "Overall title") + theme(
  panel.background = element_rect(fill = "lemonchiffon1",
                                colour = "lemonchiffon1",
                                size = 0.5, linetype = "solid")))

The violin plot above demonstrates a distribution that appears to be fairly normal with some slight skew, a median BMI that is just under 30, and an interquartile range that is between BMI values slightly greater than 25 and slightly less than 35. The normal QQ plot is consistent with what was seen in the histogram and on the violin plot, which is a mostly normal distribution with right skew.

Skew Assessment:

In order to ensure that the right skew that was noticed was not so much that a transformation will be required for further analysis of this data, a numerical skew assessment was completed. In this analysis, nonparametric skew is calculated by taking the difference between the mean and the median, and dividing it by the standard deviation. This particular type of skew is based off of Pearson’s notion of median skewness, with values falling between -1 and +1 for any distribution. Generally, when this measure of skew is used, values greater than + 0.2 indicate fairly substantial right skew, while values below -0.2 indicate fairly substantial left skew. If the skewness value for bmi is between -0.2 and +0.2, I will consider the the data to be normally distributed.

Stroke_Analysis_Train |>
  summarize(skew = round_half_up((mean(bmi) - median(bmi))/sd(bmi), 3)) |> 
  kbl(caption = " Numerical Skew Assessment for BMI values", digits = 5) |> 
  kable_styling(font_size = 20) |> 
  kable_paper("hover", 
              full_width = F)
Numerical Skew Assessment for BMI values
skew
0.158

The value derived by taking the difference between the mean and the median, and dividing it by the standard deviation provided a skewness value of .158, which is not large enough to consider there to be substantial right skew in the Stroke_Analysis_Train data set. For this reason, the distribution of BMI in this set will be considered to be normal.

6.2 Numerical Summary of the outcome of interest (BMI) for Stroke_Analysis_Train

The following code provides a numerical summary for the Stroke_Analysis_Train data set, which will be used to create a linear regression model.

favstats(~bmi, data = Stroke_Analysis_Train) |> 
  kbl(caption = "BMI Numerical Summary", digits = 5) |> 
  kable_styling(font_size = 20) |> 
  kable_paper("hover", full_width = F)
BMI Numerical Summary
min Q1 median Q3 max mean sd n missing
14.1 25.4 29 33.9 49.9 30.00474 6.3684 2299 0

6.3 Numerical Summaries of the predictors for Stroke_Analysis_Train

The following code is present for the purpose of providing a numerical summary of the intended predictors of the linear model that will be built using Stroke_Analysis_Train data. A total of four quantitative and two categorical variables will be included.

Stroke_Analysis_Train |> select(-id, -bmi ) |> 
  mosaic::inspect() 

categorical variables:  
            name  class levels    n missing
1      work_type factor      3 2299       0
2 Residence_type factor      2 2299       0
3 smoking_status factor      3 2299       0
4         stroke factor      2 2299       0
                                   distribution
1 Private (64.8%), Self-employed (18.7%) ...   
2 Urban (51.2%), Rural (48.8%)                 
3 never smoked (52.6%) ...                     
4 No Stroke (94.5%), Stroke (5.5%)             

quantitative variables:  
               name   class   min    Q1 median     Q3    max
1               age numeric 13.00 35.00  50.00  64.00  82.00
2 avg_glucose_level numeric 55.22 77.36  91.92 116.07 271.74
       mean       sd    n missing
1  49.66464 18.34327 2299       0
2 108.29753 47.70805 2299       0

6.4 Scatterplot Matrices

Two scatter plot matrices were created for the purpose of visualizing the relationship between each of the predictors and the outcome of interest being studied for Stroke_Analysis_Train using the code below:

spm1A <- Stroke_Analysis_Train |> 
  select(age, avg_glucose_level, Residence_type, bmi) 

spm2A <- Stroke_Analysis_Train |>
select(stroke, smoking_status, work_type, bmi)

ggpairs(spm1A, title = "Scatterplot Matrix",
        lower = list(combo = wrap("facethist", bins = 20))) + 
  theme(plot.background = element_rect(fill = "azure2")) + 
  theme( panel.background = element_rect(fill = "lemonchiffon1",
                                colour = "lemonchiffon1",
                                size = 0.5, 
                                linetype = "solid")) +
  labs(title = "Scatterplot Matrix",
       subtitle = 
         "Comparison of age, average blood glucose level, residence type, and BMI")

ggpairs(spm2A, title = "Scatterplot Matrix",
        lower = list(combo = wrap("facethist", bins = 20)), 
        ggplot2:: aes()) +  theme(plot.background = element_rect(fill = "azure2")) + 
  theme(panel.background = element_rect(fill = "lemonchiffon1",
                                colour = "lemonchiffon1",
                                size = 0.5, linetype = "solid")) +
  labs(title = "Scatterplot Matrix",
       subtitle = 
         "Comparison of stroke status, smoking status, employment category, and BMI ")

It can be seen above that scatter plots between quantitative variables demonstrate relative heteroscedasticity. Furthermore, the distribution of bmi and age appear to be relatively normal. The distribution of avg_glucose_level shows some right skew. avg_glucose_lavel and age both appear to have weak positive correlations with bmi, with the relationship between age and bmi being considerably weak (correlation of 0.059). avg_glucose_level and age also appear to have a weak positive relationship, with a correlation of 0.240.

In terms of qualitative values, there does not appear to be a strong relationship between any of the quantitative predictors and qualitative predictors. For example, residence_type and age. The following numerical summary demonstrates that the residence type is similar among the different age groups:

mosaic::favstats(age ~ Residence_type, data = Stroke_Analysis_Train) |> 
  kbl(caption = "Age Numerical Summary by Residence type", digits = 5) |> 
  kable_styling(font_size = 20) |> 
  kable_paper("hover", full_width = F)
Age Numerical Summary by Residence type
Residence_type min Q1 median Q3 max mean sd n missing
Rural 13 34 50 63 82 49.14248 18.35837 1123 0
Urban 13 36 50 65 82 50.16327 18.32275 1176 0

A similar relationship between predictors as that which is seen in the above table was also seen for avg_glucose_level and Residence_type, in which there was a fairly even distribution between the average blood glucose levels of people living in both urban and rural areas.

mosaic::favstats(avg_glucose_level ~ Residence_type, data = Stroke_Analysis_Train) |>
  kbl(caption = "Average Blood Glucose Level Numerical Summary by Residence type", digits = 5) |> 
  kable_styling(font_size = 20) |> 
  kable_paper("hover", 
              full_width = F)
Average Blood Glucose Level Numerical Summary by Residence type
Residence_type min Q1 median Q3 max mean sd n missing
Rural 55.27 77.670 92.87 116.290 271.74 108.4020 47.62479 1123 0
Urban 55.22 76.905 91.20 116.025 267.61 108.1978 47.80747 1176 0

Because smoking has been demonstrated to lead to life-ending medical conditions, age and smoking status were compared. However, it did not appear that there was an category of smoking that was vastly different in its age distribution. Minimum and maximum values were remarkably similar and interquartile ranges were not immensely varied.

mosaic::favstats(age ~ smoking_status, data = Stroke_Analysis_Train) |> 
  kbl(caption = "Age Numerical Summary by Smoking Status", digits = 5) |> 
  kable_styling(font_size = 20) |> 
  kable_paper("hover", 
              full_width = F)
Age Numerical Summary by Smoking Status
smoking_status min Q1 median Q3 max mean sd n missing
never smoked 13 33 48 63 82 48.11332 19.03359 1209 0
formerly smoked 13 43 58 70 82 55.66025 17.28137 571 0
smokes 16 32 47 58 82 46.68208 16.28111 519 0

6.4.1 Discussion of colinearity

As discussed above, avg_glucose_level and age appear to have a weak positive relationship, with a correlation of 0.240. For the purpose of this study, a pearson correlation value under 0.3 will be considered to be weak enough to not consider there to be an issue with collinearity between predictors. Thus, there does not appear to be any issues with collinearity that may affect the model that will be made using data from the Stroke_Analysis_Train data set.

6.5 Checking to see if a Transformation is Necessary:

In order to determine whether a linearizing transformation is necessary, a Box Cox plot method was employed. For this analysis, a linear model that will later be described as the “Big model” was used. The purpose of constructing a box-Cox plot is to sift through Tukey’s ladder of power transformations and determine which transformation is most suitable for linearizing the relationship between outcome and predictors.

par(bg = "azure2")

model_temp <- lm( bmi~ work_type + 
                    Residence_type + 
                    stroke + 
                    smoking_status + 
                    age + 
                    avg_glucose_level, data = Stroke_Analysis_Train)

boxCox(model_temp)

The boxCox approach is used in conjunction with Tukey’s ladder of power transformations:

Power (λ) -2 -1 -1/2 0 1/2 1 2
Transformation 1/y2 1/y \(\frac{1}{\sqrt[]{y}}\) log(y) \(\sqrt{y}\) y y\(^3\)

In order to show the numeric value of λ displayed above, the powerTransform function from the car package was used:

powerTransform(model_temp)
Estimated transformation parameter 
       Y1 
-0.122814

The value of γ derived from the powerTransform function is close to zero, indicating that a natural logarithmic transformation of bmi may be useful for this data (using log, which is used for natural logs in R). The following code is present for the purpose of displaying the non-transformed data alongside a possible logarithmic transformation in the format of a scatter plot of avg_glucose_level (the key predictor) and bmi (the key outcome):

p1 <- ggplot(Stroke_Analysis_Train, aes(x = avg_glucose_level, y = log(bmi))) +
  geom_point() +
  geom_smooth(method = "loess", formula = y ~ x, se = FALSE) + 
  geom_smooth(method = "lm", col = "red", formula = y ~ x, se = FALSE) + theme(
  panel.background = element_rect(fill = "lemonchiffon1",
                                colour = "lemonchiffon1",
                                size = 0.5, linetype = "solid")) +
  labs(title = "log(BMI) vs Avg. Glucose Level",
       x = "Average Blood Glucose Level",
       y = "log(BMI)" )

p2 <- ggplot(Stroke_Analysis_Train, aes(x = avg_glucose_level, y = bmi)) +
  geom_point() +
  geom_smooth(method = "loess", formula = y ~ x, se = FALSE) + 
  geom_smooth(method = "lm", col = "red", formula = y ~ x, se = FALSE) + theme(
  panel.background = element_rect(fill = "lemonchiffon1",
                                colour = "lemonchiffon1",
                                size = 0.5, linetype = "solid")) + 
  labs(title = "BMI vs Avg. Glucose Level",
       x = "Average Blood Glucose Level",
       y = "log(BMI)" )

(p1)+ theme(plot.background = element_rect(fill = "azure2")) + p2 + 
  theme(plot.background = element_rect(fill = "azure2"))

The above comparison demonstrates that when the natural log of bmi is used, a slightly more linear relationship is observed. For this reason, the transformation determined by the Box-Cox method will be kept for this analysis.

7 The Big Model

The following code is present for the purpose of fitting a linear model to predict the natural log of bmi given information on avg_glucose_level, work_type, Residence_type, stroke, smoking_status and age. a summary of each of the coefficients is also provided. This model is titled Big_model to reflect that it includes many variables. Another model will be built called Small_model that only includes one predictor (the key predictor, avg_glucose_level)

Big_model <- lm(log(bmi) ~ avg_glucose_level +
                  work_type + 
                  Residence_type + 
                  stroke + 
                  smoking_status + 
                  age, 
                data = Stroke_Analysis_Train)

summary(Big_model)

Call:
lm(formula = log(bmi) ~ avg_glucose_level + work_type + Residence_type + 
    stroke + smoking_status + age, data = Stroke_Analysis_Train)

Residuals:
     Min       1Q   Median       3Q      Max 
-0.72432 -0.14660 -0.01194  0.13966  0.57687 

Coefficients:
                                Estimate Std. Error t value Pr(>|t|)
(Intercept)                    3.286e+00  1.898e-02 173.084  < 2e-16
avg_glucose_level              5.733e-04  9.389e-05   6.106  1.2e-09
work_typePrivate              -7.823e-03  1.196e-02  -0.654   0.5132
work_typeSelf-employed        -1.747e-02  1.482e-02  -1.179   0.2386
Residence_typeUrban            3.990e-03  8.662e-03   0.461   0.6451
strokeStroke                  -1.491e-02  1.965e-02  -0.759   0.4480
smoking_statusformerly smoked  1.614e-02  1.068e-02   1.511   0.1308
smoking_statussmokes           1.722e-02  1.089e-02   1.581   0.1141
age                            6.215e-04  2.654e-04   2.341   0.0193
                                 
(Intercept)                   ***
avg_glucose_level             ***
work_typePrivate                 
work_typeSelf-employed           
Residence_typeUrban              
strokeStroke                     
smoking_statusformerly smoked    
smoking_statussmokes             
age                           *  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.2072 on 2290 degrees of freedom
Multiple R-squared:  0.02459,   Adjusted R-squared:  0.02118 
F-statistic: 7.216 on 8 and 2290 DF,  p-value: 1.732e-09

7.1 Effect Sizes: Coefficient Estimates

The following code is present for the purpose of displaying all coefficients included in the Big_model linear model. Included is information on the size, magnitude, and estimated effect sizes with 90% confidence intervals for each coefficient.

tidy(Big_model, conf.int = TRUE, conf.level = 0.90) |> 
  select(term, estimate, std.error, conf.low, conf.high, p.value) |> 
  kbl(caption = "Coeficient Estimates with 90% confisence intervals", 
      digits = 5) |> kable_styling(font_size = 20) |> 
  kable_paper("hover", full_width = F)
Coeficient Estimates with 90% confisence intervals
term estimate std.error conf.low conf.high p.value
(Intercept) 3.28562 0.01898 3.25439 3.31686 0.00000
avg_glucose_level 0.00057 0.00009 0.00042 0.00073 0.00000
work_typePrivate -0.00782 0.01196 -0.02751 0.01186 0.51319
work_typeSelf-employed -0.01747 0.01482 -0.04187 0.00692 0.23855
Residence_typeUrban 0.00399 0.00866 -0.01026 0.01824 0.64510
strokeStroke -0.01491 0.01965 -0.04725 0.01742 0.44798
smoking_statusformerly smoked 0.01614 0.01068 -0.00143 0.03371 0.13084
smoking_statussmokes 0.01722 0.01089 -0.00071 0.03515 0.11408
age 0.00062 0.00027 0.00018 0.00106 0.01930

It can be seen in the above table that none of the coefficients for any of the variables are particularly large, suggesting that none are strong predictors of the natural log of bmi. Furthermore, many of the 90% confidence intervals calculated for the coefficients had ranges that included zero, suggesting that the sample size was not large enough to determine whether or not they had a positive or negative effect on the natural log of bmi. Taking into account only the point estimates, the model suggests that being self employed, working for a private company, and having suffered a stroke are associated with lower ln(bmi) values while all other variables positively correlate with ln(bmi).

7.1.1 Visualization of Estimates

Find below a plot summary containing the 90% confidence intervals of each of the coefficients displayed in the table above. In the following plot, the intercept is not displayed as it is several orders of magnitude greater than the largest predictor coefficient.

td <- tidy(Big_model, conf.int = TRUE, conf.level = 0.90) |> 
  filter(term != "(Intercept)")
ggplot(td, aes(x = term,
               y = estimate, 
               col = term)) +
  geom_point(size = 2) +
  geom_errorbar(aes(ymin = conf.low, ymax = conf.high), width = 0.3) +
  ylim(-0.05,0.04) +  theme(legend.key=element_rect(colour="azure2")) +
  theme(axis.text.x=element_text(angle=90,hjust=1)) +
  labs(title ="Big Model Estimates",
    subtitle = "With 90% Confidence intervals",
       x="",
       y="Estimate for coefficient (with 90% CI)") +
    guides(x =  guide_axis(angle = 90)) + 
  theme(panel.background = element_rect(fill = "lemonchiffon1",
                                colour = "lemonchiffon1",
                                size = 0.5, linetype = "solid")) + 
  theme(plot.background = element_rect(fill = "azure2"))

7.2 Displaying the Equation For the Big Model

Using the point estimates calculated for each coefficient, an equation can be written for the Big_model. The following code displays this equation:

extract_eq(Big_model, use_coefs = TRUE, coef_digits = 5,
           terms_per_line = 2, wrap = TRUE, ital_vars = TRUE)

\[ \begin{aligned} \widehat{log(bmi)} &= 3.28562 + 0.00057(avg\_glucose\_level)\ - \\ &\quad 0.00782(work\_type_{Private}) - 0.01747(work\_type_{Self-employed})\ + \\ &\quad 0.00399(Residence\_type_{Urban}) - 0.01491(stroke_{Stroke})\ + \\ &\quad 0.01614(smoking\_status_{formerly\ smoked}) + 0.01722(smoking\_status_{smokes})\ + \\ &\quad 0.00062(age) \end{aligned} \]

The above equation indicates the following:

  • As the average blood glucose level for a patient increases by 1 mg/dl, the natural log of the BMI for that patient should increase by a factor of 0.00057.

  • If a patient is privately employed, the natural log of the BMI for that patient should decrease by a factor of 0.00782.

  • If a patient is self employed, the natural log of the BMI for that patient should decrease by a factor of 0.01747.

  • If a patient lives in an urban environment as opposed to a rural environment, the natural log of the BMI for that patient should increase by a factor of 0.00399.

  • If a patient has suffered a stroke, the natural log of the BMI for that patient should decrease by a factor of 0.01491.

  • If a patient has formerly smoked or is a current smoker, that patient’s natural log for BMI should increase by a factor of 0.01614 or 0.01722 respectively.

  • For every additional year that a patient is alive, that patient’s natural log for BMI should increase by a factor of .00062.

Please note that each of the above bullet points assume that all other variables are held constant

8 The Smaller Model

The following code is present for the purpose of fitting a linear model to predict the natural log of bmi given information only on avg_glucose_level, which is the key predictor for this study. This model is termed Small_model, as it only uses one variable to predict the natural log of bmi.

Small_model <- lm(log(bmi) ~ avg_glucose_level, 
                  data = Stroke_Analysis_Train)

summary(Small_model)

Call:
lm(formula = log(bmi) ~ avg_glucose_level, data = Stroke_Analysis_Train)

Residuals:
     Min       1Q   Median       3Q      Max 
-0.71701 -0.14609 -0.00984  0.14059  0.56056 

Coefficients:
                   Estimate Std. Error t value Pr(>|t|)    
(Intercept)       3.312e+00  1.073e-02 308.696  < 2e-16 ***
avg_glucose_level 6.176e-04  9.068e-05   6.811 1.24e-11 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.2074 on 2297 degrees of freedom
Multiple R-squared:  0.01979,   Adjusted R-squared:  0.01937 
F-statistic: 46.39 on 1 and 2297 DF,  p-value: 1.236e-11

8.1 Effect Sizes: Coefficient Estimates

The following code is present for the purpose of displaying the intercept point estimate as well as the point estimate for the coefficient included in the small linear model for the key predictor (avg_glucose_level). Included is information on the size, magnitude, and estimated effect sizes with 90% confidence intervals for the intercept and coefficient.

tidy(Small_model, conf.int = TRUE, conf.level = 0.90) |> 
  select(term, estimate, std.error, conf.low, conf.high, p.value) |>
  kbl(caption = "Coeficient Estimates with 90% confisence intervals", 
      digits = 5) |> 
  kable_styling(font_size = 20) |> 
  kable_paper("hover", full_width = F)
Coeficient Estimates with 90% confisence intervals
term estimate std.error conf.low conf.high p.value
(Intercept) 3.31248 0.01073 3.29483 3.33014 0
avg_glucose_level 0.00062 0.00009 0.00047 0.00077 0

It can be seen above that the small model estimates a very small positive effect on the natural log of the bmi value. It is notable that the 90% confidence interval is comprised of only positive values, indicating that with the sample size used to determine this estimate, at an alpha level of 0.10, the estimate is statistically meaningfully positive.

8.1.1 Plotting the estimate:

The following plot displays the 90% confidence interval for the avg_glucose_level coefficient displayed in the table above. In the following plot, the intercept for the Small_model is not displayed as it is several orders of magnitude greater than the predictor coefficient.

td2 <- tidy(Small_model, conf.int = TRUE, conf.level = 0.90) |> 
  filter(term != "(Intercept)")
ggplot(td2, aes(x = term, y = estimate, col = term)) +
  geom_point(size = 2) +
  geom_errorbar(aes(ymin = conf.low, ymax = conf.high), width = 0.3) + 
  ylim(-0.00,0.001) +  theme(legend.key=element_rect(colour="azure2")) +
  theme(axis.text.x=element_text(angle=90,hjust=1)) +
  labs(title ="Big Model Estimates",
    subtitle = "With 90% Confidence intervals",
       x="",
       y="Estimate for coefficient (with 90% CI)") +
    guides(x =  guide_axis(angle = 90)) + 
  theme(panel.background = element_rect(fill = "lemonchiffon1",
                                colour = "lemonchiffon1",
                                size = 0.5, linetype = "solid")) + 
  theme(plot.background = element_rect(fill = "azure2"))

8.2 Displaying the Equation For the Small Model

Using the point estimate calculated the coefficient as well as the intercept, an equation can be written for the Small_model. The following code displays this equation:

extract_eq(Small_model, use_coefs = TRUE, coef_digits = 5,
           terms_per_line = 2, wrap = TRUE, ital_vars = TRUE)

\[ \begin{aligned} \widehat{log(bmi)} &= 3.31248 + 0.00062(avg\_glucose\_level) \end{aligned} \]

The above equation indicates the following:

  • As the average blood glucose level for a patient increases by 1 mg/dl, the natural log of the BMI for that patient should increase by a factor of 0.00062.

  • The Y intercept, or the value for BMI that is predicted to be present if a patient had a blood glucose level of 0 mg/dl, is 3.31248. While this number is useful for the purpose of plotting the model, it is practically impossible for a living person to have a blood glucose level of 0 mg/dl or a BMI below 8.

9 In sample comparison

9.1 Quality of fit

The following code is present for the purpose of displaying a table that may be used to compare the Big_model to the Small_model. This table includes information related to the number of observations used to create each model, the degrees of freedom, the Akaike information criterion (AIC) and the Bayesian information criterion (BIC).

temp_a <- glance(Big_model) |> 
  select(-logLik) |>
  round(digits = 5) |>
  mutate(modelname = "Big Model")

temp_b <- glance(Small_model) |>
  select(-logLik) |>
  round(digits = 5) |>
  mutate(modelname = "Small Model")

training_model_comparison <- bind_rows(temp_a, temp_b) |>
  select(modelname, nobs, df, AIC, BIC, everything())


training_model_comparison |> kbl(caption = "Comparison of training models", 
                                 digits = 5) |> 
  kable_styling(font_size = 20) |> 
  kable_paper("hover", full_width = F)
Comparison of training models
modelname nobs df AIC BIC r.squared adj.r.squared sigma statistic p.value deviance df.residual
Big Model 2299 8 -702.5240 -645.1217 0.02459 0.02118 0.20719 7.21593 0 98.30501 2290
Small Model 2299 1 -705.2513 -688.0306 0.01979 0.01937 0.20738 46.38534 0 98.78821 2297

The above table demonstrates that both the Small_model and the Big_model are based off of the same number of observations, which makes sense as the Stroke_Analysis_Train data frame was composed of only complete cases. Although the AIC is similar between the two models, it is smaller in the big model and suggests that the big model is better fit than the smaller model. Furthermore, the BIC for the Big_model was smaller than that of the Small_model, also suggesting that the big model demonstrates better fit than the smaller model. This makes sense because BIC tends to be smaller for models with more predictors. Finally, the R squared value for the Big_model was larger than that of the Small_model, indicating that the big model is able to explain more of the variation seen in the natural log of bmi than the smaller model. This was to be expected, as the Big_model contains the same variable as the Small_model in addition to more variables and R squared values tend to favor models with more predictors.

9.2 Assessing Assumptions:

In order to exhibit the differences between observed and fitted response values for each of the models, residual plots were created

9.2.1 Residual Plots for the Big Model

The following code is present for the purpose of displaying residual plots for the big model, including a residuals vs fitted plot, a normal qq plot for the residuals, a scale location plot for the residuals, and a residuals vs leverage plot.

par(bg = "azure2")

par(mfrow = c(2,2)); plot(Big_model); par(mfrow = c(1,1))

It can be observed above that the residuals vs. fitted plot showed linearity. Additionally, the normal QQ plot exhibited a normal distribution among the residual values for the big model, and the scale-location plot displayed heteroscedasticity and thus constant variance. Finally, there did not appear to be any exceptionally large Cook’s distance values on the residuals vs. leverage plot, indicating that none of the data points had seriously disproportionate influence on the model. Given the results for each of these plots, it doesn’t appear that any of the assumptions for multiple linear regression are present (linearity,independence, equal variance, or normality ).

9.2.2 Residual Plots for the Small Model

The following code is present for the purpose of displaying residual plots for the small model, including a residuals vs fitted plot, a normal qq plot for the residuals, a scale location plot for the residuals, and a residuals vs leverage plot.

par(bg = "azure2")

par(mfrow = c(2,2)); plot(Small_model); par(mfrow = c(1,1))

In the above residual plots for Small_model, it can be seen that, similar to Big_model, the residuals vs. fitted plot showed linearity, the normal QQ plot exhibited a normal distribution among the residual values for the big model, and the scale-location plot displayed heteroscedasticity and thus constant variance. Additionally, there did not appear to be any exceptionally large Cook’s distance values on the residuals vs. leverage plot, indicating that none of the data points had seriously disproportionate influence on the model. Given the results for each of these plots, it doesn’t appear that any of the assumptions for multiple linear regression are present (linearity,independence, equal variance, or normality ).

9.2.3 Assessing the Collinearity of the Big Model

Due to the fact that the big model had multiple predictors, the potential impact of collinearity should be explored. The variance inflation factor (VIF) is commonly used to quantify the impact of collinearity for linear models. The following code is present for the purpose of displaying the Variance Inflation Factor using the vif function in the car package.

car::vif(Big_model) |> 
  kbl(caption = "Variance Inflation Factor Analysis of the Big Model", 
      digits = 5) |> 
  kable_styling(font_size = 20) |> 
  kable_paper("hover", full_width = F)
Variance Inflation Factor Analysis of the Big Model
GVIF Df GVIF^(1/(2*Df))
avg_glucose_level 1.07418 1 1.03643
work_type 1.11427 2 1.02742
Residence_type 1.00391 1 1.00195
stroke 1.07913 1 1.03881
smoking_status 1.04167 2 1.01026
age 1.26919 1 1.12659

The above VIF table does not exhibit any values for VIF above 1.27. Generally, VIF values greater than or equal to 5 are considered to indicate a serious issue with multicollinearity, as this would correspond to R squared values at or above 0.8. No VIF values above are near this, and for this reason there is is not considered to be any notable multicollinearity issues in the Big_model.

Because the Small_model only uses one predictor, it is not necessary to complete a multicollinearity assessment for that model.

9.3 Comparing the models

Based on the above information comparing the Big_model with the Small_model, it appears that although neither of the models violated any important regression assumptions, the fit quality is better for the Big_model than it is for the Small_model. This is due to a number of important factors. The Big_model showed smaller values for both AIC and BIC as well as a larger R squared value than the Small_model. This suggests that the fit quality is higher in the Big_model and that it is able to explain more of the variation in bmi values. Overall, it appears that the big model is a better model to use to predict bmi values than the small model.

Although both of the models were shown to have some predictive value, it should be noted that none of the coefficients for any of the variables used to attempt to predict bmi were very large, and thus it should be considered that none of the variables used in either of the models had an exceptionally strong influence on the bmi value predicted.

10 Model Validation

In this section, the two models created in the previous section will be used to predict BMI using data from Stroke_Analysis_Test.

10.1 Calculating the prediction errors

10.1.1 Big Model: Back-Transformation and Calculating Fits/Residuals

Using the augment function from within the broom package, fitted and residual values were determined and stored as bmi_fit and bmi_res respectively for the big model on the Stroke_Analysis_Test data. Because a natural logarithmic transformation was used in the development of each of the linear models, a back-transformation is necessary in order to proceed with this function. To back-transform the data, Euler’s number was raised to a power the the value for bmi.

aug_big_model <- augment(Big_model, newdata = Stroke_Analysis_Test) |> 
  mutate(mod_name = "Big",
         bmi_fit = exp(1)^(.fitted),
         bmi_res = bmi - bmi_fit) |>
  select(id, mod_name, bmi, bmi_fit, bmi_res, everything())

head(aug_big_model,4) |> 
  kbl(caption = "Residual and Fitted Values Calculated For the Big Model", 
      digits = 5) |> 
  kable_styling(font_size = 20) |> 
  kable_paper("hover", full_width = F)
Residual and Fitted Values Calculated For the Big Model
id mod_name bmi bmi_fit bmi_res age avg_glucose_level work_type Residence_type smoking_status stroke .fitted .resid
1665 Big 24.0 30.02899 -6.02899 79 174.12 Self-employed Rural never smoked Stroke 3.40216 -0.22411
53882 Big 27.4 28.47604 -1.07604 74 70.09 Private Rural never smoked Stroke 3.34906 -0.03852
58202 Big 30.9 29.37938 1.52062 50 167.41 Self-employed Rural never smoked Stroke 3.38029 0.05046
56112 Big 37.5 30.99208 6.50792 64 191.61 Private Urban smokes Stroke 3.43373 0.19061

10.1.2 Small Model: Back-Transformation and Calculating Fits/Residuals

Using the augment function from within the broom package, fitted and residual values were determined and stored as bmi_fit and bmi_res respectively for the small model on the Stroke_Analysis_Test data. Once again, a back transformation was required to remove the natural log that had been applied to bmi values earlier.

aug_small_model <- augment(Small_model, newdata = Stroke_Analysis_Test) |> 
  mutate(mod_name = "Small",
         bmi_fit = exp(1)^(.fitted),
         bmi_res = bmi - bmi_fit) |>
  select(id, mod_name, bmi, bmi_fit, bmi_res, everything())

head(aug_small_model,4) |> 
  kbl(caption = "Residual and Fitted Values Calculated For the Small Model", 
      digits = 5) |> 
  kable_styling(font_size = 20) |> 
  kable_paper("hover", full_width = F)
Residual and Fitted Values Calculated For the Small Model
id mod_name bmi bmi_fit bmi_res age avg_glucose_level work_type Residence_type smoking_status stroke .fitted .resid
1665 Small 24.0 30.56992 -6.56992 79 174.12 Self-employed Rural never smoked Stroke 3.42002 -0.24196
53882 Small 27.4 28.66765 -1.26765 74 70.09 Private Rural never smoked Stroke 3.35577 -0.04523
58202 Small 30.9 30.44350 0.45650 50 167.41 Self-employed Rural never smoked Stroke 3.41587 0.01488
56112 Small 37.5 30.90191 6.59809 64 191.61 Private Urban smokes Stroke 3.43082 0.19352

10.1.3 Combining the results

The following code is present for the purpose of creating a combined tibble containing residual and fit information from both aug_small_model and aug_big_model.

test_comp <- union(aug_big_model, aug_small_model) |>
  arrange(id, mod_name)

head(test_comp, 6) |> 
  kbl(caption = 
        "Table of Residual and Fitted Values Calculated For the Big and Small Model", 
      digits = 5) |> 
  kable_styling(font_size = 20) |> kable_paper("hover", full_width = F)
Table of Residual and Fitted Values Calculated For the Big and Small Model
id mod_name bmi bmi_fit bmi_res age avg_glucose_level work_type Residence_type smoking_status stroke .fitted .resid
10138 Big 24.8 28.97181 -4.17181 41 74.85 Private Urban formerly smoked No Stroke 3.36632 -0.15548
10138 Small 24.8 28.75204 -3.95204 41 74.85 Private Urban formerly smoked No Stroke 3.35871 -0.14787
10159 Big 31.6 28.91873 2.68127 41 99.80 Private Urban never smoked No Stroke 3.36449 0.08867
10159 Small 31.6 29.19851 2.40149 41 99.80 Private Urban never smoked No Stroke 3.37412 0.07904
10245 Big 35.8 28.39331 7.40669 54 77.52 Self-employed Rural never smoked No Stroke 3.34615 0.23179
10245 Small 35.8 28.79949 7.00051 54 77.52 Self-employed Rural never smoked No Stroke 3.36036 0.21759

10.2 Visualizing the Predictions

The following code is present for the purpose of visualizing the predicted and observed values for each model side-by-side.

ggplot(test_comp, aes(x = bmi_fit, y = bmi)) +
  geom_point() +
  geom_abline(slope = 1, intercept = 0, lty = "dashed") + 
  geom_smooth(method = "loess", col = "blue", se = FALSE, formula = y ~ x) +
  facet_wrap( ~ mod_name, labeller = "label_both") +
  labs(x = "Predicted BMI",
       y = "Observed BMI",
       title = "Observed vs. Predicted BMI",
       subtitle = "Comparison of Big and Small Model Residual And Fitted Values",
       caption = "Dashed line represents where Observed = Predicted") + 
  theme(
  panel.background = element_rect(fill = "lemonchiffon1",
                                colour = "lemonchiffon1",
                                size = 0.5, linetype = "solid")) + 
  theme(plot.background = element_rect(fill = "azure2"))

It can be seen in the plots above that both the Big_model and the Small_model predict the values for BMI conservatively and do not predict extreme values for BMI. It may also be seen that the general shape of both models are similar, with the Big_model demonstrating a slightly flatter Loess smooth line. It appears from the scatter plots that the Big_model predicted more BMI values below 28 than did the Small_model and that neither model predicted a BMI value above 33.

Summarizing the Errors

The following code is present for the purpose of displaying a table containing a summary of the prediction errors for each model. This includes the mean absolute prediction error, or MAPE and the square root of the mean squared prediction error, or RMSPE.

test_comp |>
  group_by(mod_name) |>
  dplyr::summarise(n = n(),
            MAPE = mean(abs(bmi_res)), 
            RMSPE = sqrt(mean(bmi^2)),
            max_error = max(abs(bmi))) |> 
  kbl(caption = "Error Summary", digits = 13) |> 
  kable_styling(font_size = 20) |> 
  kable_paper("hover", full_width = F)
Error Summary
mod_name n MAPE RMSPE max_error
Big 986 4.922390 30.6549 49.8
Small 986 4.947876 30.6549 49.8

It can be noted in the table above that both the big and the small model demonstrated similar values for MAPE and RMSPE, with the only notable difference being in the MAPE, which was a slightly smaller for the big model. This indicates some possibility that the big model is more suited to answer the research question.

10.3 Identifying the largest error.

The following code is present for the purpose of determining the observation in which the Big_model was least able to accurately predict the value for bmi and the observation in which the Small_model was least able to accurately predict the value for bmi.

temp1 <- aug_big_model |>
  filter(abs(bmi_res) == max(abs(bmi_res)))

temp2 <- aug_small_model |>
  filter(abs(bmi_res) == max(abs(bmi_res)))

bind_rows(temp1, temp2) |> 
  kbl(digits = 5) |> kable_styling(font_size = 20) |> 
  kable_paper("hover", full_width = F)
id mod_name bmi bmi_fit bmi_res age avg_glucose_level work_type Residence_type smoking_status stroke .fitted .resid
59872 Big 49.3 28.43877 20.86123 38 80.82 Private Rural never smoked No Stroke 3.34775 0.55017
18498 Small 49.8 29.27036 20.52964 44 103.78 Private Rural formerly smoked No Stroke 3.37658 0.53144

It appears in the table above that the greatest errors for both models involved an underestimation of the BMI value and that both models incorrectly predicted the BMI of a patient living in a rural area and working for a private company whose actual BMI was close to 49 and whose age is below 45. Neither of these patients had suffered a stroke.

10.4 Validated R squared values

The following code is present for the purpose of calculating the R squared values for the Big_model and the Small_model.

#Big model: 

cor(aug_big_model$bmi, aug_big_model$bmi_fit)^2
[1] 0.04260902
#Small model: 

cor(aug_small_model$bmi, aug_small_model$bmi_fit)^2
[1] 0.04210552
Model R-Squared value
Big Model 0.04260902
Small Model 0.04210552

It appears that around 4% of the variation in the BMI within the Stroke_Analysis_Test data is explained by both of the models. However, it appears that a slightly greater percent of variation in BMI is explained by the big model. This in conjunction with its slightly smaller value for Mean Absolute Percentage Error (MAPE) shows the big model to be a better predictor for BMI (albeit very marginally).

10.5 Comparing the Models

Although the models were similarly in their ability to predict values for Stroke_Analysis_Test, it appears that the Big_model is a better predictive model. This is because it exhibited a smaller mean absolute prediction error than the Small_model as well as an R squared value that was greater than that of the Small_model. These indicate that the Big_model was better able to predict the BMI value of a patient included in the Stroke_Analysis_Test data set and that the variation in BMI is better explained by this model.

11 Discussion

The Chosen Model

It was decided that the Big_model should be the chosen model for this study. This is because it was demonstrated to be better fitted to the training sample (Stroke_Analysis_train), and better able to predict values of BMI for the test sample (Stroke_Analysis_Test). It was found to have both a smaller AIC and a smaller BIC value in the in-sample comparison than the Small_model, indicating that it has a greater quality of fit. This was further demonstrated in residual vs. fitted plots. Furthermore, the Big_model was found to have a smaller MAPE value than the Small_model when it was used to predict the values of Stroke_Analysis_Test, showing that it had the ability to predict BMI with less error. Finally, the Big_model had a greater R squared value than the Small_model, demonstrating that it was able to explain a larger amount of the variation in bmi in the Stroke_Analysis_Test data. Importantly, the Big_model also did not violate any assumptions for linear regression, as it was seen in the residual plots that it demonstrated linearity, independence, equal variance, and normality in its distribution. Had it violated any of these key assumptions its utility as a linear model would be severely compromised.

Answering the research question

The research question for this analysis was:

How effectively can the body mass index of a hospital patient who has been evaluated for a stroke be predicted by the average blood glucose level of the patient that was recorded in their EHR and how is the quality of prediction changed when adjusting for age, employment type, residence type, stroke history, and smoking status?

To answer this question, both the Small_model and the Big_model need to be taken into consideration as the Small_model involves uses only the average blood glucose level as a predictor and the Big_model adjusts for age, employment type, residence type, stroke history, and smoking status. In the in-sample comparison, the Big_model was found to have been fit more closely to the values in the training data set than the Small_model, demonstrating that the model that was built with adjustment made for age, employment type, residence type, stroke history, and smoking status had smaller residual values (between predicted and observed BMI values) for the data it is built on than one that only uses the average blood glucose level as a predictor. This suggests that there is a possibility that BMI is able to be more effectively predicted when adjustments for age, employment type, residence type, stroke history, and smoking status are made, but further testing is required to make that statement confidently. For this reason, a model validation study was completed, in which the Big_model and the Small_model were compared in their ability to predict BMI values using data that they were not trained on.

The model validation study applied each of the models to an unfamiliar data set of 986 new observations (named Stroke_Analysis_Test). Fitted and residual values for each model were calculated and compared using numeric summaries and plots. It was observed that the mean absolute prediction error for the model which adjusted for age, employment type, residence type, stroke history, and smoking status was lower than the MAPE for the model that only incorporated average blood glucose level as a predictor for BMI. Furthermore it was observed that the model in which age, employment type, residence type, stroke history, and smoking status were adjusted for had a higher R squared value than the model that only incorporated average blood glucose level as a predictor for BMI. Both of these observations further substantiated the previously mentioned possibility that the quality of prediction of BMI increased when average blood glucose level was used as a predictor and adjustments are made for age, employment type, residence type, stroke history, and smoking status.

While it appeared that the quality of BMI prediction increased with the adjustments mentioned above, it is important to discuss the quality of prediction in general for both models. In the Small_model, where only average blood glucose was used as a predictor, it was found that average blood glucose demonstrated a very weak positive association with BMI, with an expected increase in the natural log of BMI of 0.00062 for every additional 1 mg/dl of blood glucose for in patient. This means that the smaller model predicts that an extremely large increase or decrease in average blood glucose is required to create even a modest change in bmi. Given this small demonstrated influence it cannot be stated confidently that average blood glucose as stated in a patient’s EHR is an effective predictor of that patient’s BMI.

In the Big_model, none of the coefficients for any of the variables used in the linear model had a value greater than 0.0147, indicating that a only a very large change (in many cases an unrealistically high change) in one of the quantitative variables would be expected to inspire a large change in the predicted BMI value of a patient. Importantly, a number of the coefficients for the variables used in this model had 90% confidence intervals that contained zero, indicating that the sample size of the data set used to create the model was not large enough to establish a positive or negative correlation between these variables and BMI with respect to an alpha level of .10. None of the categorical variables were demonstrated to have the ability to change the predicted BMI value of a patient by a factor that would alter their status as underweight, healthy, overweight, or obese as per CDC guidance. For this reason, it can be said that although a marginally better quality of prediction is observed when predicting BMI using average blood glucose level and adjustments are made for age, employment type, residence type, stroke history, and smoking status, the quality of prediction is still very low. R squared values in the validation study demonstrated that both the Big_model and the Small_model were each only able to explain around 4% of the change in BMI within the Stroke_Analysis_Test data set.

Given each of the points made above, it appears that the research question is best answered as follows:

The body mass index of a hospital patient who has been evaluated for a stroke cannot be effectively predicted by the average blood glucose level that was recorded in their EHR. However, slightly more accurate predictions of BMI for hospital patients who have been evaluated for stroke can be calculated by using average blood glucose level of the patient as a predictor and adjusting for age, employment type, residence type, stroke history, and smoking status of the patient.

My pre-analysis expectations were that the the body mass index of a hospital patient who has been evaluated for stroke status would be able to be effectively predicted by the average blood glucose level of that patient and that adjusting for age, employment type, residence type, stroke history, and smoking status would improve the predictive quality. Although I had correctly anticipated that the inclusion of additional variables would improve the predictive quality, the weak predictive ability of the average blood glucose level was not at all expected.

There are a number of limitations to this study. First, the sample size used to train the model was limited, which in turn limited the ability of both of the models to be to reflect the true distribution of data for patients who have been evaluated for stroke. Second, the models created were strictly linear and thus it is possible that although average blood glucose cannot be used in a linear model to predict BMI adequately, it may be a useful predictor when incorporated into a different type of regression model (such as a parabolic or cubic model).

Next Steps

In looking to create a prediction model for BMI using electronic health record data, there are a number of steps that may be taken in the future to create a more comprehensive model with a higher quality of BMI prediction. One item to consider is the amount of observations used to train the model. Because in 2022 the use of electronic health record systems is fairly ubiquitous, it is not unrealistic to consider that data from far more than 2299 patients could be used to create a predictive model for BMI. Additionally, more variables may be considered in making this model, specifically those that are known to modulate the mass of a person, such as participation in exercise or physical disability.

Additionally, one may consider using an approach to predicting BMI that does not involve a linear model, but rather using cubic or quadratic models to account for the possibility that different variables exert different effects on a patient’s BMI based on where that person lies on the distribution of BMI values.

Finally, further studies looking to establish a predictive model for BMI may consider looking beyond only patients who have been evaluated for a stroke as per their electronic health records. This may involve looking at populations that have been evaluated for other diseases or looking at a broader spectrum of hospital patients with less specific constraints. It is possible that people who have been evaluated for a stroke do not have particularly strong correlations between BMI and the variables examined in this study, but that other populations do.

Reflection

Reflecting on my approach to this study knowing what I do now, there are a few of things that had I known at the beginning of the study, I would have done differently.

Before I completed this project, I was not aware of how I could display the confidence intervals for the coefficients of a linear model created using ggplot2. In completing this project I found it to be an incredibly useful way to visualize the influence of variables in a linear model and a useful tool for explaining an equation. If I had understood how to do this at the beginning of the project I would have created a plot of coefficients and confidence intervals before writing any explanation of my model so that I could have more quickly and effectively described its equation and anticipated the effects of each of its variables.

Additionally, when I began working on this project, I did not have a strong understanding of the car package and how it can be used to analyze the the Variance Inflation Factor in table form to assess collinearity of a multiple linear regression model. When I began the project, I was planning on only using pearson correlations and scatter plot matrices to analyze collinearity, however, if I had been better acquainted with the vif function, I may have chosen additional variables for my big model, as I would more closely be able to examine whether or not there was collinearity between any of my predictors.

Session Information

sessioninfo::session_info()
─ Session info ─────────────────────────────────────────────────────
 setting  value
 version  R version 4.2.1 (2022-06-23)
 os       macOS Big Sur ... 10.16
 system   x86_64, darwin17.0
 ui       X11
 language (EN)
 collate  en_US.UTF-8
 ctype    en_US.UTF-8
 tz       America/New_York
 date     2022-12-19
 pandoc   2.19.2 @ /Applications/RStudio.app/Contents/MacOS/quarto/bin/tools/ (via rmarkdown)

─ Packages ─────────────────────────────────────────────────────────
 package        * version    date (UTC) lib source
 abind            1.4-5      2016-07-21 [1] CRAN (R 4.2.0)
 assertthat       0.2.1      2019-03-21 [1] CRAN (R 4.2.0)
 backports        1.4.1      2021-12-13 [1] CRAN (R 4.2.0)
 base64enc        0.1-3      2015-07-28 [1] CRAN (R 4.2.0)
 bookdown         0.30       2022-11-09 [1] CRAN (R 4.2.0)
 broom          * 1.0.1      2022-08-29 [1] CRAN (R 4.2.0)
 bslib            0.4.1      2022-11-02 [1] CRAN (R 4.2.0)
 cachem           1.0.6      2021-08-19 [1] CRAN (R 4.2.0)
 car            * 3.1-1      2022-10-19 [1] CRAN (R 4.2.0)
 carData        * 3.0-5      2022-01-06 [1] CRAN (R 4.2.0)
 cellranger       1.1.0      2016-07-27 [1] CRAN (R 4.2.0)
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 cli              3.4.1      2022-09-23 [1] CRAN (R 4.2.0)
 cluster          2.1.4      2022-08-22 [1] CRAN (R 4.2.0)
 cmprsk           2.2-11     2022-01-06 [1] CRAN (R 4.2.0)
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 crayon           1.5.2      2022-09-29 [1] CRAN (R 4.2.0)
 data.table       1.14.6     2022-11-16 [1] CRAN (R 4.2.0)
 DBI              1.1.3      2022-06-18 [1] CRAN (R 4.2.0)
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 deldir           1.0-6      2021-10-23 [1] CRAN (R 4.2.0)
 digest           0.6.31     2022-12-11 [1] CRAN (R 4.2.1)
 distributional   0.3.1      2022-09-02 [1] CRAN (R 4.2.0)
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 ellipsis         0.3.2      2021-04-29 [1] CRAN (R 4.2.0)
 Epi            * 2.47       2022-06-26 [1] CRAN (R 4.2.0)
 equatiomatic   * 0.3.1      2022-01-30 [1] CRAN (R 4.2.0)
 etm              1.1.1      2020-09-08 [1] CRAN (R 4.2.0)
 evaluate         0.18       2022-11-07 [1] CRAN (R 4.2.0)
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 fastmap          1.1.0      2021-01-25 [1] CRAN (R 4.2.0)
 forcats        * 0.5.2      2022-08-19 [1] CRAN (R 4.2.0)
 foreign          0.8-83     2022-09-28 [1] CRAN (R 4.2.0)
 Formula          1.2-4      2020-10-16 [1] CRAN (R 4.2.0)
 fs               1.5.2      2021-12-08 [1] CRAN (R 4.2.0)
 gargle           1.2.1      2022-09-08 [1] CRAN (R 4.2.0)
 generics         0.1.3      2022-07-05 [1] CRAN (R 4.2.0)
 GGally         * 2.1.2      2021-06-21 [1] CRAN (R 4.2.0)
 ggdist         * 3.2.0      2022-07-19 [1] CRAN (R 4.2.0)
 ggforce        * 0.4.1      2022-10-04 [1] CRAN (R 4.2.0)
 ggformula      * 0.10.2     2022-09-01 [1] CRAN (R 4.2.0)
 gghalves       * 0.1.4      2022-11-20 [1] CRAN (R 4.2.0)
 ggmosaic       * 0.3.4      2022-12-10 [1] Github (haleyjeppson/ggmosaic@fb42e7b)
 ggplot2        * 3.4.0      2022-11-04 [1] CRAN (R 4.2.0)
 ggrepel          0.9.2      2022-11-06 [1] CRAN (R 4.2.0)
 ggridges         0.5.4      2022-09-26 [1] CRAN (R 4.2.0)
 ggstance       * 0.3.5      2020-12-17 [1] CRAN (R 4.2.0)
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 Hmisc            4.7-1      2022-08-15 [1] CRAN (R 4.2.0)
 hms              1.1.2      2022-08-19 [1] CRAN (R 4.2.0)
 htmlTable        2.4.1      2022-07-07 [1] CRAN (R 4.2.0)
 htmltools        0.5.4      2022-12-07 [1] CRAN (R 4.2.0)
 htmlwidgets      1.5.4      2021-09-08 [1] CRAN (R 4.2.0)
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 httr             1.4.4      2022-08-17 [1] CRAN (R 4.2.0)
 interp           1.1-3      2022-07-13 [1] CRAN (R 4.2.0)
 janitor        * 2.1.0      2021-01-05 [1] CRAN (R 4.2.0)
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 later            1.3.0      2021-08-18 [1] CRAN (R 4.2.0)
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 lazyeval         0.2.2      2019-03-15 [1] CRAN (R 4.2.0)
 lifecycle        1.0.3      2022-10-07 [1] CRAN (R 4.2.0)
 lubridate        1.9.0      2022-11-06 [1] CRAN (R 4.2.0)
 magrittr       * 2.0.3      2022-03-30 [1] CRAN (R 4.2.0)
 MASS             7.3-58.1   2022-08-03 [1] CRAN (R 4.2.0)
 Matrix         * 1.4-1      2022-03-23 [1] CRAN (R 4.2.1)
 mgcv             1.8-41     2022-10-21 [1] CRAN (R 4.2.0)
 mime             0.12       2021-09-28 [1] CRAN (R 4.2.0)
 modelr           0.1.10     2022-11-11 [1] CRAN (R 4.2.1)
 modelsummary   * 1.2.0      2022-11-26 [1] CRAN (R 4.2.0)
 mosaic         * 1.8.4.2    2022-09-20 [1] CRAN (R 4.2.0)
 mosaicCore       0.9.2.1    2022-09-22 [1] CRAN (R 4.2.0)
 mosaicData     * 0.20.3     2022-09-01 [1] CRAN (R 4.2.0)
 munsell          0.5.0      2018-06-12 [1] CRAN (R 4.2.0)
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 numDeriv         2016.8-1.1 2019-06-06 [1] CRAN (R 4.2.0)
 patchwork      * 1.1.2      2022-08-19 [1] CRAN (R 4.2.0)
 pillar           1.8.1      2022-08-19 [1] CRAN (R 4.2.0)
 pkgconfig        2.0.3      2019-09-22 [1] CRAN (R 4.2.0)
 plotly           4.10.1     2022-11-07 [1] CRAN (R 4.2.0)
 plyr             1.8.8      2022-11-11 [1] CRAN (R 4.2.1)
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 [1] /Library/Frameworks/R.framework/Versions/4.2/Resources/library

────────────────────────────────────────────────────────────────────
---
title: "Evaluation of BMI Prediction Models Among Patients Who Have Undergone a Stroke Evaluation"
author: "Benjamin Heifetz"
date: "`r Sys.Date()`"
output: 
  rmdformats::readthedown:
    highlight: kate
    number_sections: TRUE
    code_folding: show
    code_download: TRUE
---

## Preliminary information {.unnumbered}

Because patients who are monitored for stroke status during hospital visits are often watched closely and put through a myriad of tests, they may have exceptionally rich EHR data and be useful clinical research subjects. The purpose of this study is to use EHR data pulled from patients who had been evaluated for stroke to create a useful model to predict BMI. 

# Setup and Data Ingest

## Setup

The packages and parameters used in this code chunk are present in order to generate a legible report.

```{r setup}

library(knitr)
library(rmdformats)
library(rmarkdown)

options(max.print="100")
knitr::opts_chunk$set(echo = TRUE)
knitr::opts_chunk$set(comment = NA)
options(width = 70)
```

## Loading Necessary R Packages

Each of the following packages were necessary for the analysis and/or graphical display of information used in this project.

```{r load_packages, message = FALSE, warning = FALSE}
setwd(dirname(getwd()))
source("Love-boost.R")
library(janitor)
library(Epi)
library(kableExtra)
library(GGally)
library(ggforce)
library(ggstance)
library(modelsummary)
library(ggdist)
library(gghalves)
library(ggmosaic)
library(car)
library(tidyr)
library(magrittr)
library(mosaic)
library(equatiomatic)
library(simputation) 
library(patchwork)
library(broom)
library(naniar)
library(tidyverse)



theme_set(theme_bw())

```

## Data ingest

The following code is present for the purpose of ingesting the raw data, which is a .csv file. Additionally, data types were standardized in this step by converting necessary character variables into factors and retaining data for `bmi` as numeric. The `id` variable is maintained as a character variable here as it is just a unique numeric identifier for each patient. It will only be used for identification purposes in subsequent analyses, not as a quantitative or categorical variable that affects the outcome of interest.

```{r, warning=FALSE}
stroke_raw <- read.csv("STROKEDATA.csv") |> 
   mutate(across(where(is.character), as.factor)) |>
  mutate(id = as.character(id))  |> 
  mutate(across(where(is.integer), as.factor)) |> 
  mutate(bmi = as.character(bmi)) |> 
  mutate(bmi = as.numeric(bmi))  


```

# Cleaning the Data

The following code is present for the purpose of creating a preliminary data frame in the form of a tibble that contains complete cases of each of the variables being analyzed in this project. It should be noted that only employed individuals are being considered in this study, and for that reason, two categories are omitted from the `work type` variable. This code also renames the components of the `stroke` variable and re-levels the components of the `smoking_status` variable in order to have more communicable results in subsequent analyses. Finally, a brief summary of the analytic tibble with complete cases, named `stroke_complete_cases` is provided.

```{r}

stroke_raw <- stroke_raw |> 
  filter(complete.cases(work_type)) |> 
  filter(work_type == "Govt_job" | 
           work_type == "Private" | 
           work_type == "Self-employed") |> 
  droplevels()


stroke_raw$smoking_status <- recode_factor(stroke_raw$smoking_status, 
                                             "Unknown" = "N/A")

stroke_complete_cases <- stroke_raw |>
 filter(complete.cases(id, 
                       age, 
                       work_type, 
                       Residence_type, 
                       avg_glucose_level, 
                       bmi, 
                       smoking_status, stroke),  
        bmi != "N/A", smoking_status != "N/A", gender != "Other") 
    

stroke_complete_cases$stroke <- recode_factor(stroke_complete_cases$stroke, 
                                       "0" = "No Stroke",
                                             "1" = "Stroke")


stroke_complete_cases <- stroke_complete_cases |> 
  mutate(smoking_status = fct_relevel(smoking_status, 
                                      "never smoked",
                                      "formerly smoked", 
                                      "smokes")) |> 
  droplevels()


stroke_complete_cases <- stroke_complete_cases |> 
  as_tibble(stroke_complete_cases)



glimpse(stroke_complete_cases)

```

### Checking to make sure that none of the values for `bmi` are unrealistically high or low: {.unnumbered}

The following code is present for the purpose of displaying the number of `bmi` observations above 50 (10 points above the clinical level of *morbid* obesity as per CDC guidance) and below 10 (8 points below the clinical *underweight* cutoff point as per the CDC). This allows for any abnormally large or small values in the data set to be noted so that they can be omitted.

**Checking the high-end of the BMI variable::**

```{r}

sum(stroke_complete_cases$bmi > 50) 

```

**Checking the low-end of the BMI variable:**

```{r}
sum(stroke_complete_cases$bmi < 10)
```

Although the lowest `bmi` value in the `stroke_complete_cases` data frame appears to be in a reasonable area for a low BMI per CDC guidance, there are a total of 58 recorded BMI values over 50, which is considerably above the widely accepted cutoff value for morbid obesity of 40 (as per CDC guidance). For this reason values of `bmi` over 50 will be removed from the data frame, as they are either serious anomalies or possibly improperly measured/reported values. The following code removes these values and rechecks the number of abnormal values:

```{r}

stroke_complete_cases <- stroke_complete_cases[which(stroke_complete_cases$bmi < 50),]

sum(stroke_complete_cases$bmi > 50) 
sum(stroke_complete_cases$bmi < 10)

```

## Checking the key outcome (BMI) and key predictor (Average Blood Glucose Level)

The following code is present for the purpose of displaying a numeric summary for the key outcome of the study (`bmi`) and the key predictor of the study (`avg_glucose_level`).

```{r}
df_stats(~bmi + avg_glucose_level, data = stroke_complete_cases) |> 
  kbl(caption = 
        "Numerical Summary of Key Outcome 
      (BMI) and Key Predictor (Average Blood Glucose Level)", digits = 2) |> 
  kable_styling(font_size = 20) |> 
  kable_paper("hover", full_width = F)
```

As expected, there are no missing values for either of the variables above. Additionally, it can be seen that all values for BMI are between 10 and 50 and the interquartile range of average blood glucose level is below 140, demonstrating that neither of these variables contain abnormal values by the clinical standards reported by the CDC.

## Checking the Quantitative Predictors:

For this analysis, there is only one quantitative predictor that is not the key predictor (`avg_glucose_level`), which is `age`. The following code provides a numeric summary of `age`.

```{r}

df_stats(~age, data = stroke_complete_cases) |> 
  kbl(caption = "Numerical summary of quantitative predictor (Age)", 
      digits = 2) |> kable_styling(font_size = 20) |> 
  kable_paper("hover", 
              full_width = F)

```

It can be seen above that a wide range of ages will be included in this study, including ages as low as 13 and as high as 82. The average and median ages appear to be around 50.

## Checking the Categorical Variables:

*In this section, numeric summaries each of the categorical variables are individually provided. Each of these variables are to be used as a predictor for the key outcome, `bmi`.*

### Residence type

The following code provides a numeric summary of the `Residence_type` variable:

```{r}

stroke_complete_cases |> 
  tabyl(Residence_type) |> 
  kbl(caption = "Summary of Residence Type", digits = 2) |> 
  kable_styling(font_size = 20) |> 
  kable_paper("hover", full_width = F)

```

It can be seen in the above table that there is nearly a 50/50 split between people included in this study that live in a rural environment and people included in this study that live in an urban environment.

### Stroke

The following code provides a numeric summary of the `stroke` variable:

```{r}

stroke_complete_cases |> tabyl(stroke) |> 
  kbl(caption = "Summary of Stroke Status",digits = 2) |> 
  kable_styling(font_size = 20) |> 
  kable_paper("hover", 
              full_width = F)

```

The above table demonstrates that only about five percent of the people included in this study had suffered a stroke in their lifetime. It should be noted that everybody in this study was evaluated for a stroke, however.

### Work Type

The following code provides a numeric summary of the `work_type` variable:

```{r}

stroke_complete_cases |> 
  tabyl(work_type) |> 
  kbl(caption = "Summary of Employment Category",digits = 2) |> 
  kable_styling(font_size = 20) |> 
  kable_paper("hover", full_width = F)

```

The above table demonstrates that most people included in this study are privately employed. Around 15% of participants appear to be self-employed and 19% appear to be government employees.

### Smoking Status

The following code provides a numeric summary of the `smoking_status` variable:

```{r}

stroke_complete_cases |> 
  tabyl(smoking_status) |> 
  kbl(caption = "Summary of Smoking Status", digits = 2) |> 
  kable_styling(font_size = 20) |> 
  kable_paper("hover", full_width = F)

```

It is demonstrated in the table above that close to half of the participants in this study have never smoked in their life, while the remaining portion is split fairly evenly between former smokers and current smokers. It appears that there are more former smokers than current smokers.

## Evaluation of missing cases

The following code is present for the purpose of consolidating information on all missing values in one place, such that it can be determined that the `stroke_complete_cases` tibble can be considered to be made up of only complete cases. It should be noted that any missing cases are considered to have been missing completely at random (MCAR).

```{r}

miss_var_summary(stroke_complete_cases)

```

## Printing The analytic tibble

The following code is present for the purpose of renaming the data set containing all of the six variables and printing it to show the final analytic tibble that will be used in all subsequent analyses.

```{r}

Stroke_Analysis_Data <- stroke_complete_cases |> 
  select(id, age, bmi, 
         avg_glucose_level, work_type, Residence_type, 
         smoking_status, stroke)

Stroke_Analysis_Data

```

# Codebook and Data Description

## Description of the subjects of this study

The data used for this study was taken from the electronic health records (EHRs) of 3285 persons aged 10-82 who, within their EHR, had explicitly been noted as either having suffered a stroke in their life or having not suffered a stroke in their life. All EHR data used in this study was de-identified and made public by McKinsey and Company, a consulting firm with oversight over a number of US hospital EHRs. All subjects had complete data for each of the variables listed below.

## Codebook

**Description of Variables**

The following table contains the name of each variable in the `Stroke_Analysis_Data` data set used for this analysis, the type of variable it is, and a brief description of the levels for each variable. The variable types are denoted as such:

*ID = Identification number*

*Quant = Quantitative variables*

*Binary = Two-category variables*

*X-cat = Multi-category variables) where X indicates the number of levels in the variable.*

| Variable            |  Type  | Description / Levels                                                                                                                                                                                                                                                                                                                    |
|-----------------|:---------------:|---------------------------------------|
| `id`                |   ID   | A unique identification number assigned to each participant                                                                                                                                                                                                                                                                             |
| `bmi`               | Quant  | **Outcome Variable**: Current body mass index of the patient as indicated by that patient's EHR. Body mass index is calculated by dividing weight in pounds (lb) by height in inches (in) squared and multiplying that value by a conversion factor of 703.                                                                             |
| `work_type`         | 3-Cat  | The category of work of the patient as indicated by that patient's EHR. **"Govt_job"** indicates that the patient works for the government (public sector), **"Private"** indicates that the patient is employed and works for a private company (private sector), and **"Self-employed"** indicates that the patient is self employed. |
| `stroke`            | Binary | Whether or not electronic medical records indicated that the individual had suffered a stroke at some point in their life. **"Stroke"** indicates that they suffered a stroke, **"No Stroke"** indicates that they did not.                                                                                                             |
| `smoking_status`    | 3-Cat  | The smoking status of the patient as indicated by that patient's EHR. **"never smoked"** indicates that the patient has no history of smoking, **"formerly smoked"** indicates that they have, but no longer smoke, and **"smokes"** indicates that the patient currently smokes.                                                       |
| `Residence_type`    | 2-cat  | The residence type in which the patient lives. **Urban** refers to an urban environment and **Rural** refers to a rural environment.                                                                                                                                                                                                    |
| `age`               | Quant  | The age of the patient in *years* as indicated in that patient's EHR.                                                                                                                                                                                                                                                                   |
| `avg_glucose_level` | Quant  | **Key Predictor** The average blood glucose level for the patient, measured in *milligrams per deciliter* (Mg/Dl) as indicated in that patient's EHR                                                                                                                                                       |


## Analytic Tibble

The following code is present for the purpose of printing the analytic tibble in order to show that it is, in fact, a tibble.

```{r}

Stroke_Analysis_Data

```

## Numerical Data Description:

The following code is present for the purpose of displaying a brief numeric description of each of the variables used in the `Stroke_Analysis_Data` data set.

```{r}
Hmisc::describe(Stroke_Analysis_Data) 
```

# My Research Question

Body mass index is a widely utilized reporting method for the relationship between the height and weight of a person. Values for BMI that are 20 and over 25 have been associated with [higher all-cause mortality](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4995441/), with increasing risk as the distance from the 20--25 range increases. Given the positive correlation between BMI and all-cause mortality risk, models that are able to accurately predict BMI may be useful for determining at-risk patients who may need additional care and resources in the future. Average blood glucose level is an easy to measure metric that can be determined at nearly any clinic, thus it may be advantageous to see if it can be used to predict BMI value. 


My research question is as follows:

**How effectively can the body mass index of a hospital patient who has been evaluated for a stroke be predicted by the average blood glucose level of the patient that was recorded in their EHR and how is the quality of prediction changed when adjusting for age, employment type, residence type, stroke history, and smoking status?**

# Partitioning the Data

The following code is present for the purpose of randomly segregating the data from `Stroke_Analysis_Data` into a "training" data frame containing 70% of the observations called `Stroke_Analysis_Train`, and a second "test" data frame containing the remaining 30% of the observations, called `Stroke_Analysis_Test`.

```{r}

set.seed(538) 

Stroke_Analysis_Train <- Stroke_Analysis_Data |> 
  slice_sample(prop = .70)
Stroke_Analysis_Test <- anti_join(Stroke_Analysis_Data, 
                                  Stroke_Analysis_Train, 
                                  by = "id")

dim(Stroke_Analysis_Data)

dim(Stroke_Analysis_Train)

dim(Stroke_Analysis_Test)

```

It can be seen above by the dimensions of the `Stroke_Analysis_Data`, `Stroke_Analysis_Train`, and `Stroke_Analysis_Test` data frames respectively, that each data frame has eight variables, and that 70% of the number of observations in `Stroke_Analysis_Data` are included in the `Stroke_Analysis_Train` data frame. 30% of the number of observations from `Stroke_Analysis_Data` are included in the `Stroke_Analysis_Test` data frame.

# Transforming The Outcome

In this section, the properties of the training data set (`Stroke_Analysis_Train`) will be assessed for the purpose of determining whether or not a transformation is necessary before creating a linear regression model using the data.

## Visualization of `Stroke_Analysis_Train` with no applied transformations

The following code creates a histogram of the distribution of BMI values across the `Stroke_Analysis_Train` data.

```{r, warning=FALSE}

ggplot(Stroke_Analysis_Train, aes(x = bmi)) +
    geom_histogram(bins = 75, fill = "lightsalmon", color = "coral2") +
  guides("none") +
    labs(title = "Histogram of BMI Values by Employment Category",
         x = "BMI") + theme(
  panel.background = element_rect(fill = "lemonchiffon1",
                                colour = "lemonchiffon1",
                                size = 0.5, linetype = "solid")) + 
  theme(plot.background = element_rect(fill = "azure2"))


```

It can be seen in the above histogram that the distribution of BMI within the `Stroke_Analysis_Train` data set appears to be fairly normal with some right skew.

The following code is present for the purpose of creating a box and violin plot juxtaposed with normal QQ plots for `bmi`, allowing for further visualization of the distribution of values in the `Stroke_Analysis_Train` data set.

```{r}

p1 <- ggplot(Stroke_Analysis_Train, aes(x ="",  
                                        y = bmi)) + 
  geom_violin(col = "coral2", 
              fill = "lightsalmon", 
              alpha = 0.3) +
  geom_boxplot(width = 0.3, notch = TRUE, 
               color = "coral2", fill = "azure2", ) +
  guides(fill = "none") +
  labs(title = "BMI Values",
       subtitle = "For the `Stroke_Analysis_Train` data set.",
       x = "`Stroke_Analysis_Train` data set", 
       y = "Body Mass Index") + 
  theme(plot.background = element_rect(fill = "azure2"))

p2 <- ggplot(Stroke_Analysis_Train, 
             aes(sample = bmi, 
                 col = "coral2")) +
  geom_qq(fill = "lightsalmon") +
  geom_qq_line(col = "darkturquoise") +
  guides(col = "none") +
  theme_bw() +
  labs(y = "BMI ",
       title = "Normal QQ Plot", 
       subtitle = "For the `Stroke_Analysis_Train` data set.") +
  theme(plot.background = element_rect(fill = "azure2"))


(p1 + theme(
  panel.background = element_rect(fill = "lemonchiffon1",
                                colour = "lemonchiffon1",
                                size = 0.5, linetype = "solid"))) + (p2  +
    plot_annotation(title = "Overall title") + theme(
  panel.background = element_rect(fill = "lemonchiffon1",
                                colour = "lemonchiffon1",
                                size = 0.5, linetype = "solid"))) 
```

The violin plot above demonstrates a distribution that appears to be fairly normal with some slight skew, a median BMI that is just under 30, and an interquartile range that is between BMI values slightly greater than 25 and slightly less than 35. The normal QQ plot is consistent with what was seen in the histogram and on the violin plot, which is a mostly normal distribution with right skew.

*Skew Assessment:*

In order to ensure that the right skew that was noticed was not so much that a transformation will be required for further analysis of this data, a numerical skew assessment was completed. In this analysis, nonparametric skew is calculated by taking the difference between the mean and the median, and dividing it by the standard deviation. This particular type of skew is based off of Pearson's notion of median skewness, with values falling between -1 and +1 for any distribution. Generally, when this measure of skew is used, values greater than + 0.2 indicate fairly substantial right skew, while values below -0.2 indicate fairly substantial left skew. If the skewness value for `bmi` is between -0.2 and +0.2, I will consider the the data to be normally distributed.

```{r}
Stroke_Analysis_Train |>
  summarize(skew = round_half_up((mean(bmi) - median(bmi))/sd(bmi), 3)) |> 
  kbl(caption = " Numerical Skew Assessment for BMI values", digits = 5) |> 
  kable_styling(font_size = 20) |> 
  kable_paper("hover", 
              full_width = F)
```

The value derived by taking the difference between the mean and the median, and dividing it by the standard deviation provided a skewness value of .158, which is not large enough to consider there to be substantial right skew in the `Stroke_Analysis_Train` data set. For this reason, the distribution of BMI in this set will be considered to be normal.

## Numerical Summary of the outcome of interest (BMI) for `Stroke_Analysis_Train`

The following code provides a numerical summary for the `Stroke_Analysis_Train` data set, which will be used to create a linear regression model.

```{r}

favstats(~bmi, data = Stroke_Analysis_Train) |> 
  kbl(caption = "BMI Numerical Summary", digits = 5) |> 
  kable_styling(font_size = 20) |> 
  kable_paper("hover", full_width = F)

```

## Numerical Summaries of the predictors for `Stroke_Analysis_Train`

The following code is present for the purpose of providing a numerical summary of the intended predictors of the linear model that will be built using `Stroke_Analysis_Train` data. A total of four quantitative and two categorical variables will be included.

```{r}
Stroke_Analysis_Train |> select(-id, -bmi ) |> 
  mosaic::inspect() 
```

## Scatterplot Matrices

Two scatter plot matrices were created for the purpose of visualizing the relationship between each of the predictors and the outcome of interest being studied for `Stroke_Analysis_Train` using the code below:

```{r}
spm1A <- Stroke_Analysis_Train |> 
  select(age, avg_glucose_level, Residence_type, bmi) 

spm2A <- Stroke_Analysis_Train |>
select(stroke, smoking_status, work_type, bmi)

ggpairs(spm1A, title = "Scatterplot Matrix",
        lower = list(combo = wrap("facethist", bins = 20))) + 
  theme(plot.background = element_rect(fill = "azure2")) + 
  theme( panel.background = element_rect(fill = "lemonchiffon1",
                                colour = "lemonchiffon1",
                                size = 0.5, 
                                linetype = "solid")) +
  labs(title = "Scatterplot Matrix",
       subtitle = 
         "Comparison of age, average blood glucose level, residence type, and BMI")


ggpairs(spm2A, title = "Scatterplot Matrix",
        lower = list(combo = wrap("facethist", bins = 20)), 
        ggplot2:: aes()) +  theme(plot.background = element_rect(fill = "azure2")) + 
  theme(panel.background = element_rect(fill = "lemonchiffon1",
                                colour = "lemonchiffon1",
                                size = 0.5, linetype = "solid")) +
  labs(title = "Scatterplot Matrix",
       subtitle = 
         "Comparison of stroke status, smoking status, employment category, and BMI ")
```

It can be seen above that scatter plots between quantitative variables demonstrate relative heteroscedasticity. Furthermore, the distribution of `bmi` and `age` appear to be relatively normal. The distribution of `avg_glucose_level` shows some right skew. `avg_glucose_lavel` and `age` both appear to have weak positive correlations with `bmi`, with the relationship between `age` and `bmi` being considerably weak (correlation of 0.059). `avg_glucose_level` and `age` also appear to have a weak positive relationship, with a correlation of 0.240.

In terms of qualitative values, there does not appear to be a strong relationship between any of the quantitative predictors and qualitative predictors. For example, `residence_type` and `age.` The following numerical summary demonstrates that the residence type is similar among the different age groups:

```{r}
mosaic::favstats(age ~ Residence_type, data = Stroke_Analysis_Train) |> 
  kbl(caption = "Age Numerical Summary by Residence type", digits = 5) |> 
  kable_styling(font_size = 20) |> 
  kable_paper("hover", full_width = F)
```

A similar relationship between predictors as that which is seen in the above table was also seen for `avg_glucose_level` and `Residence_type`, in which there was a fairly even distribution between the average blood glucose levels of people living in both urban and rural areas.

```{r}

mosaic::favstats(avg_glucose_level ~ Residence_type, data = Stroke_Analysis_Train) |>
  kbl(caption = "Average Blood Glucose Level Numerical Summary by Residence type", digits = 5) |> 
  kable_styling(font_size = 20) |> 
  kable_paper("hover", 
              full_width = F)

```

Because smoking has been demonstrated to lead to life-ending medical conditions, age and smoking status were compared. However, it did not appear that there was an category of smoking that was vastly different in its age distribution. Minimum and maximum values were remarkably similar and interquartile ranges were not immensely varied.

```{r}
mosaic::favstats(age ~ smoking_status, data = Stroke_Analysis_Train) |> 
  kbl(caption = "Age Numerical Summary by Smoking Status", digits = 5) |> 
  kable_styling(font_size = 20) |> 
  kable_paper("hover", 
              full_width = F)
```

### Discussion of colinearity

As discussed above, `avg_glucose_level` and `age` appear to have a weak positive relationship, with a correlation of 0.240. For the purpose of this study, a pearson correlation value under 0.3 will be considered to be weak enough to not consider there to be an issue with collinearity between predictors. Thus, there does not appear to be any issues with collinearity that may affect the model that will be made using data from the `Stroke_Analysis_Train` data set.

## Checking to see if a Transformation is Necessary:

In order to determine whether a linearizing transformation is necessary, a Box Cox plot method was employed. For this analysis, a linear model that will later be described as the "Big model" was used. The purpose of constructing a box-Cox plot is to sift through Tukey's ladder of power transformations and determine which transformation is most suitable for linearizing the relationship between outcome and predictors.

```{r}
par(bg = "azure2")

model_temp <- lm( bmi~ work_type + 
                    Residence_type + 
                    stroke + 
                    smoking_status + 
                    age + 
                    avg_glucose_level, data = Stroke_Analysis_Train)

boxCox(model_temp) 
```

The `boxCox` approach is used in conjunction with Tukey's ladder of power transformations:

|                |        |        |                        |        |            |       |       |
|---------|---------|---------|---------|---------|---------|---------|---------|
| **Power (λ)**  | **-2** | **-1** | **-1/2**               | **0**  | **1/2**    | **1** | **2** |
| Transformation | 1/y^2^ | 1/y    | $\frac{1}{\sqrt[]{y}}$ | log(y) | $\sqrt{y}$ | y     | y$^3$ |

In order to show the numeric value of λ displayed above, the `powerTransform` function from the `car` package was used:

```{r}
powerTransform(model_temp)
```

The value of γ derived from the `powerTransform` function is close to zero, indicating that a natural logarithmic transformation of `bmi` may be useful for this data (using `log`, which is used for natural logs in R). The following code is present for the purpose of displaying the non-transformed data alongside a possible logarithmic transformation in the format of a scatter plot of `avg_glucose_level` (the key predictor) and `bmi` (the key outcome):

```{r}

p1 <- ggplot(Stroke_Analysis_Train, aes(x = avg_glucose_level, y = log(bmi))) +
  geom_point() +
  geom_smooth(method = "loess", formula = y ~ x, se = FALSE) + 
  geom_smooth(method = "lm", col = "red", formula = y ~ x, se = FALSE) + theme(
  panel.background = element_rect(fill = "lemonchiffon1",
                                colour = "lemonchiffon1",
                                size = 0.5, linetype = "solid")) +
  labs(title = "log(BMI) vs Avg. Glucose Level",
       x = "Average Blood Glucose Level",
       y = "log(BMI)" )

p2 <- ggplot(Stroke_Analysis_Train, aes(x = avg_glucose_level, y = bmi)) +
  geom_point() +
  geom_smooth(method = "loess", formula = y ~ x, se = FALSE) + 
  geom_smooth(method = "lm", col = "red", formula = y ~ x, se = FALSE) + theme(
  panel.background = element_rect(fill = "lemonchiffon1",
                                colour = "lemonchiffon1",
                                size = 0.5, linetype = "solid")) + 
  labs(title = "BMI vs Avg. Glucose Level",
       x = "Average Blood Glucose Level",
       y = "log(BMI)" )

(p1)+ theme(plot.background = element_rect(fill = "azure2")) + p2 + 
  theme(plot.background = element_rect(fill = "azure2"))


```

The above comparison demonstrates that when the natural log of bmi is used, a slightly more linear relationship is observed. For this reason, the transformation determined by the Box-Cox method will be kept for this analysis.

# The Big Model

The following code is present for the purpose of fitting a linear model to predict the natural log of `bmi` given information on `avg_glucose_level`, `work_type`, `Residence_type`, `stroke`, `smoking_status` and `age`. a summary of each of the coefficients is also provided. **This model is titled `Big_model` to reflect that it includes many variables. Another model will be built called `Small_model` that only includes one predictor (the key predictor, `avg_glucose_level`)**

```{r}

Big_model <- lm(log(bmi) ~ avg_glucose_level +
                  work_type + 
                  Residence_type + 
                  stroke + 
                  smoking_status + 
                  age, 
                data = Stroke_Analysis_Train)

summary(Big_model)

```

## Effect Sizes: Coefficient Estimates

The following code is present for the purpose of displaying all coefficients included in the `Big_model` linear model. Included is information on the size, magnitude, and estimated effect sizes with 90% confidence intervals for each coefficient.

```{r}
tidy(Big_model, conf.int = TRUE, conf.level = 0.90) |> 
  select(term, estimate, std.error, conf.low, conf.high, p.value) |> 
  kbl(caption = "Coeficient Estimates with 90% confisence intervals", 
      digits = 5) |> kable_styling(font_size = 20) |> 
  kable_paper("hover", full_width = F)
  
```

It can be seen in the above table that none of the coefficients for any of the variables are particularly large, suggesting that none are strong predictors of the natural log of `bmi`. Furthermore, many of the 90% confidence intervals calculated for the coefficients had ranges that included zero, suggesting that the sample size was not large enough to determine whether or not they had a positive or negative effect on the natural log of `bmi`. Taking into account only the point estimates, the model suggests that being self employed, working for a private company, and having suffered a stroke are associated with lower ln(`bmi`) values while all other variables positively correlate with ln(`bmi`).

### Visualization of Estimates

Find below a plot summary containing the 90% confidence intervals of each of the coefficients displayed in the table above. In the following plot, the intercept is not displayed as it is several orders of magnitude greater than the largest predictor coefficient.

```{r, warning=FALSE}
td <- tidy(Big_model, conf.int = TRUE, conf.level = 0.90) |> 
  filter(term != "(Intercept)")
ggplot(td, aes(x = term,
               y = estimate, 
               col = term)) +
  geom_point(size = 2) +
  geom_errorbar(aes(ymin = conf.low, ymax = conf.high), width = 0.3) +
  ylim(-0.05,0.04) +  theme(legend.key=element_rect(colour="azure2")) +
  theme(axis.text.x=element_text(angle=90,hjust=1)) +
  labs(title ="Big Model Estimates",
    subtitle = "With 90% Confidence intervals",
       x="",
       y="Estimate for coefficient (with 90% CI)") +
    guides(x =  guide_axis(angle = 90)) + 
  theme(panel.background = element_rect(fill = "lemonchiffon1",
                                colour = "lemonchiffon1",
                                size = 0.5, linetype = "solid")) + 
  theme(plot.background = element_rect(fill = "azure2"))

```

## Displaying the Equation For the Big Model

Using the point estimates calculated for each coefficient, an equation can be written for the `Big_model`. The following code displays this equation:

```{r}
extract_eq(Big_model, use_coefs = TRUE, coef_digits = 5,
           terms_per_line = 2, wrap = TRUE, ital_vars = TRUE)
```

The above equation indicates the following:

-   As the average blood glucose level for a patient increases by 1 mg/dl, the natural log of the BMI for that patient should increase by a factor of 0.00057.

-   If a patient is privately employed, the natural log of the BMI for that patient should decrease by a factor of 0.00782.

-   If a patient is self employed, the natural log of the BMI for that patient should decrease by a factor of 0.01747.

-   If a patient lives in an urban environment as opposed to a rural environment, the natural log of the BMI for that patient should increase by a factor of 0.00399.

-   If a patient has suffered a stroke, the natural log of the BMI for that patient should decrease by a factor of 0.01491.

-   If a patient has formerly smoked or is a current smoker, that patient's natural log for BMI should increase by a factor of 0.01614 or 0.01722 respectively.

-   For every additional year that a patient is alive, that patient's natural log for BMI should increase by a factor of .00062.

*Please note that each of the above bullet points assume that all other variables are held constant*

# The Smaller Model

The following code is present for the purpose of fitting a linear model to predict the natural log of `bmi` given information only on `avg_glucose_level`, which is the key predictor for this study. This model is termed `Small_model`, as it only uses one variable to predict the natural log of `bmi`.

```{r}

Small_model <- lm(log(bmi) ~ avg_glucose_level, 
                  data = Stroke_Analysis_Train)

summary(Small_model)

```

## Effect Sizes: Coefficient Estimates

The following code is present for the purpose of displaying the intercept point estimate as well as the point estimate for the coefficient included in the small linear model for the key predictor (`avg_glucose_level`). Included is information on the size, magnitude, and estimated effect sizes with 90% confidence intervals for the intercept and coefficient.

```{r}
tidy(Small_model, conf.int = TRUE, conf.level = 0.90) |> 
  select(term, estimate, std.error, conf.low, conf.high, p.value) |>
  kbl(caption = "Coeficient Estimates with 90% confisence intervals", 
      digits = 5) |> 
  kable_styling(font_size = 20) |> 
  kable_paper("hover", full_width = F)
  
```

It can be seen above that the small model estimates a very small positive effect on the natural log of the `bmi` value. It is notable that the 90% confidence interval is comprised of only positive values, indicating that with the sample size used to determine this estimate, at an alpha level of 0.10, the estimate is statistically meaningfully positive.

### Plotting the estimate:

The following plot displays the 90% confidence interval for the `avg_glucose_level` coefficient displayed in the table above. In the following plot, the intercept for the `Small_model` is not displayed as it is several orders of magnitude greater than the predictor coefficient.

```{r, warning = FALSE}
td2 <- tidy(Small_model, conf.int = TRUE, conf.level = 0.90) |> 
  filter(term != "(Intercept)")
ggplot(td2, aes(x = term, y = estimate, col = term)) +
  geom_point(size = 2) +
  geom_errorbar(aes(ymin = conf.low, ymax = conf.high), width = 0.3) + 
  ylim(-0.00,0.001) +  theme(legend.key=element_rect(colour="azure2")) +
  theme(axis.text.x=element_text(angle=90,hjust=1)) +
  labs(title ="Big Model Estimates",
    subtitle = "With 90% Confidence intervals",
       x="",
       y="Estimate for coefficient (with 90% CI)") +
    guides(x =  guide_axis(angle = 90)) + 
  theme(panel.background = element_rect(fill = "lemonchiffon1",
                                colour = "lemonchiffon1",
                                size = 0.5, linetype = "solid")) + 
  theme(plot.background = element_rect(fill = "azure2"))
```

## Displaying the Equation For the Small Model

Using the point estimate calculated the coefficient as well as the intercept, an equation can be written for the `Small_model`. The following code displays this equation:

```{r}
extract_eq(Small_model, use_coefs = TRUE, coef_digits = 5,
           terms_per_line = 2, wrap = TRUE, ital_vars = TRUE)

```

The above equation indicates the following:

-   As the average blood glucose level for a patient increases by 1 mg/dl, the natural log of the BMI for that patient should increase by a factor of 0.00062.

-   The Y intercept, or the value for BMI that is predicted to be present if a patient had a blood glucose level of 0 mg/dl, is 3.31248. *While this number is useful for the purpose of plotting the model, it is practically impossible for a living person to have a blood glucose level of 0 mg/dl or a BMI below 8.*

# In sample comparison

## Quality of fit

The following code is present for the purpose of displaying a table that may be used to compare the `Big_model` to the `Small_model`. This table includes information related to the number of observations used to create each model, the degrees of freedom, the Akaike information criterion (AIC) and the Bayesian information criterion (BIC). 

```{r}
temp_a <- glance(Big_model) |> 
  select(-logLik) |>
  round(digits = 5) |>
  mutate(modelname = "Big Model")

temp_b <- glance(Small_model) |>
  select(-logLik) |>
  round(digits = 5) |>
  mutate(modelname = "Small Model")

training_model_comparison <- bind_rows(temp_a, temp_b) |>
  select(modelname, nobs, df, AIC, BIC, everything())


training_model_comparison |> kbl(caption = "Comparison of training models", 
                                 digits = 5) |> 
  kable_styling(font_size = 20) |> 
  kable_paper("hover", full_width = F)


```

The above table demonstrates that both the `Small_model` and the `Big_model` are based off of the same number of observations, which makes sense as the `Stroke_Analysis_Train` data frame was composed of only complete cases. Although the AIC is similar between the two models, it is smaller in the big model and suggests that the big model is better fit than the smaller model. Furthermore, the BIC for the `Big_model` was smaller than that of the `Small_model`, also suggesting that the big model demonstrates better fit than the smaller model. This makes sense because BIC tends to be smaller for models with more predictors. Finally, the R squared value for the `Big_model` was larger than that of the `Small_model`, indicating that the big model is able to explain more of the variation seen in the natural log of `bmi` than the smaller model. This was to be expected, as the `Big_model` contains the same variable as the `Small_model` in addition to more variables and R squared values tend to favor models with more predictors. 


## Assessing Assumptions:

**In order to exhibit the differences between observed and fitted response values for each of the models, residual plots were created** 

### Residual Plots for the Big Model

The following code is present for the purpose of displaying residual plots for the big model, including a residuals vs fitted plot, a normal qq plot for the residuals, a scale location plot for the residuals, and a residuals vs leverage plot. 

```{r, fig.height=8}

par(bg = "azure2")

par(mfrow = c(2,2)); plot(Big_model); par(mfrow = c(1,1))

```

It can be observed above that the *residuals vs. fitted* plot showed linearity. Additionally, the *normal QQ plot* exhibited a normal distribution among the residual values for the big model, and the *scale-location plot* displayed heteroscedasticity and thus constant variance. Finally, there did not appear to be any exceptionally large Cook's distance values on the *residuals vs. leverage* plot, indicating that none of the data points had seriously disproportionate influence on the model. Given the results for each of these plots, it doesn't appear that any of the assumptions for multiple linear regression are present (linearity,independence, equal variance, or normality ). 


### Residual Plots for the Small Model

The following code is present for the purpose of displaying residual plots for the small model, including a residuals vs fitted plot, a normal qq plot for the residuals, a scale location plot for the residuals, and a residuals vs leverage plot. 
```{r, fig.height= 8}

par(bg = "azure2")

par(mfrow = c(2,2)); plot(Small_model); par(mfrow = c(1,1))

```

In the above residual plots for `Small_model`, it can be seen that, similar to `Big_model`, the *residuals vs. fitted* plot showed linearity, the *normal QQ plot* exhibited a normal distribution among the residual values for the big model, and the *scale-location plot* displayed heteroscedasticity and thus constant variance. Additionally, there did not appear to be any exceptionally large Cook's distance values on the *residuals vs. leverage* plot, indicating that none of the data points had seriously disproportionate influence on the model. Given the results for each of these plots, it doesn't appear that any of the assumptions for multiple linear regression are present (linearity,independence, equal variance, or normality ).

### Assessing the Collinearity of the Big Model

Due to the fact that the big model had multiple predictors, the potential impact of collinearity should be explored. The variance inflation factor (VIF) is commonly used to quantify the impact of collinearity for linear models. The following code is present for the purpose of displaying the Variance Inflation Factor using the `vif` function in the `car` package. 

```{r}

car::vif(Big_model) |> 
  kbl(caption = "Variance Inflation Factor Analysis of the Big Model", 
      digits = 5) |> 
  kable_styling(font_size = 20) |> 
  kable_paper("hover", full_width = F)
```

The above VIF table does not exhibit any values for VIF above 1.27. Generally, VIF values greater than or equal to 5 are considered to indicate a serious issue with multicollinearity, as this would correspond to R squared values at or above 0.8. No VIF values above are near this, and for this reason there is is not considered to be any notable multicollinearity issues in the `Big_model`. 

Because the `Small_model` only uses one predictor, it is not necessary to complete a multicollinearity assessment for that model. 


## Comparing the models

Based on the above information comparing the `Big_model` with the `Small_model`, it appears that although neither of the models violated any important regression assumptions, the fit quality is better for the `Big_model` than it is for the `Small_model`. This is due to a number of important factors. The `Big_model` showed smaller values for both AIC and BIC as well as a larger R squared value than the `Small_model`. This suggests that the fit quality is higher in the `Big_model` and that it is able to explain more of the variation in `bmi` values. **Overall, it appears that the big model is a better model to use to predict bmi values than the small model.**

Although both of the models were shown to have some predictive value, it should be noted that none of the coefficients for any of the variables used to attempt to predict `bmi` were very large, and thus it should be considered that none of the variables used in either of the models had an exceptionally strong influence on the `bmi` value predicted. 

# Model Validation

In this section, the two models created in the previous section will be used to predict BMI using data from `Stroke_Analysis_Test`.  

## Calculating the prediction errors

### Big Model: Back-Transformation and Calculating Fits/Residuals

Using the `augment` function from within the `broom` package, fitted and residual values were determined and stored as `bmi_fit` and `bmi_res` respectively for the big model on the `Stroke_Analysis_Test` data. Because a natural logarithmic transformation was used in the development of each of the linear models, a back-transformation is necessary in order to proceed with this function. To back-transform the data, Euler's number was raised to a power the the value for `bmi`. 

```{r}
aug_big_model <- augment(Big_model, newdata = Stroke_Analysis_Test) |> 
  mutate(mod_name = "Big",
         bmi_fit = exp(1)^(.fitted),
         bmi_res = bmi - bmi_fit) |>
  select(id, mod_name, bmi, bmi_fit, bmi_res, everything())

head(aug_big_model,4) |> 
  kbl(caption = "Residual and Fitted Values Calculated For the Big Model", 
      digits = 5) |> 
  kable_styling(font_size = 20) |> 
  kable_paper("hover", full_width = F)

```

### Small Model: Back-Transformation and Calculating Fits/Residuals 

Using the `augment` function from within the `broom` package, fitted and residual values were determined and stored as `bmi_fit` and `bmi_res` respectively for the small model on the `Stroke_Analysis_Test` data. Once again, a back transformation was required to remove the natural log that had been applied to `bmi` values earlier. 


```{r}
aug_small_model <- augment(Small_model, newdata = Stroke_Analysis_Test) |> 
  mutate(mod_name = "Small",
         bmi_fit = exp(1)^(.fitted),
         bmi_res = bmi - bmi_fit) |>
  select(id, mod_name, bmi, bmi_fit, bmi_res, everything())

head(aug_small_model,4) |> 
  kbl(caption = "Residual and Fitted Values Calculated For the Small Model", 
      digits = 5) |> 
  kable_styling(font_size = 20) |> 
  kable_paper("hover", full_width = F)
```

### Combining the results

The following code is present for the purpose of creating a combined tibble containing residual and fit information from both `aug_small_model` and `aug_big_model`.

```{r}
test_comp <- union(aug_big_model, aug_small_model) |>
  arrange(id, mod_name)

head(test_comp, 6) |> 
  kbl(caption = 
        "Table of Residual and Fitted Values Calculated For the Big and Small Model", 
      digits = 5) |> 
  kable_styling(font_size = 20) |> kable_paper("hover", full_width = F)

```

## Visualizing the Predictions

The following code is present for the purpose of visualizing the predicted and observed values for each model side-by-side.

```{r}

ggplot(test_comp, aes(x = bmi_fit, y = bmi)) +
  geom_point() +
  geom_abline(slope = 1, intercept = 0, lty = "dashed") + 
  geom_smooth(method = "loess", col = "blue", se = FALSE, formula = y ~ x) +
  facet_wrap( ~ mod_name, labeller = "label_both") +
  labs(x = "Predicted BMI",
       y = "Observed BMI",
       title = "Observed vs. Predicted BMI",
       subtitle = "Comparison of Big and Small Model Residual And Fitted Values",
       caption = "Dashed line represents where Observed = Predicted") + 
  theme(
  panel.background = element_rect(fill = "lemonchiffon1",
                                colour = "lemonchiffon1",
                                size = 0.5, linetype = "solid")) + 
  theme(plot.background = element_rect(fill = "azure2"))

```

It can be seen in the plots above that both the `Big_model` and the `Small_model` predict the values for BMI conservatively and do not predict extreme values for BMI. It may also be seen that the general shape of both models are similar, with the `Big_model` demonstrating a slightly flatter Loess smooth line. It appears from the scatter plots that the `Big_model` predicted more BMI values below 28 than did the `Small_model` and that neither model predicted a BMI value above 33. 

## Summarizing the Errors {.unnumbered}

The following code is present for the purpose of displaying a table containing a summary of the prediction errors for each model. This includes the mean absolute prediction error, or MAPE and the square root of the mean squared prediction error, or RMSPE.

```{r}

test_comp |>
  group_by(mod_name) |>
  dplyr::summarise(n = n(),
            MAPE = mean(abs(bmi_res)), 
            RMSPE = sqrt(mean(bmi^2)),
            max_error = max(abs(bmi))) |> 
  kbl(caption = "Error Summary", digits = 13) |> 
  kable_styling(font_size = 20) |> 
  kable_paper("hover", full_width = F)

```

It can be noted in the table above that both the big and the small model demonstrated similar values for MAPE and RMSPE, with the only notable difference being in the MAPE, which was a slightly smaller for the big model. This indicates some possibility that the big model is more suited to answer the research question.  

## Identifying the largest error.

The following code is present for the purpose of determining the observation in which the `Big_model` was least able to accurately predict the value for `bmi` and the observation in which the `Small_model` was least able to accurately predict the value for `bmi`. 

```{r}
temp1 <- aug_big_model |>
  filter(abs(bmi_res) == max(abs(bmi_res)))

temp2 <- aug_small_model |>
  filter(abs(bmi_res) == max(abs(bmi_res)))

bind_rows(temp1, temp2) |> 
  kbl(digits = 5) |> kable_styling(font_size = 20) |> 
  kable_paper("hover", full_width = F)

```

It appears in the table above that the greatest errors for both models involved an underestimation of the BMI value and that both models incorrectly predicted the BMI of a patient living in a rural area and working for a private company whose actual BMI was close to 49 and whose age is below 45. Neither of these patients had suffered a stroke. 


## Validated R squared values

The following code is present for the purpose of calculating the R squared values for the `Big_model` and the `Small_model`. 

```{r}

#Big model: 

cor(aug_big_model$bmi, aug_big_model$bmi_fit)^2

#Small model: 

cor(aug_small_model$bmi, aug_small_model$bmi_fit)^2

```

|             |                       |
|-------------|-----------------------|
| ***Model*** | ***R-Squared value*** |
| Big Model   | 0.04260902            |
| Small Model | 0.04210552            |

It appears that around 4% of the variation in the BMI within the Stroke_Analysis_Test data is explained by both of the models. However, it appears that a slightly greater percent of variation in BMI is explained by the big model. This in conjunction with its slightly smaller value for Mean Absolute Percentage Error (MAPE) shows the big model to be a better predictor for BMI (albeit very marginally).

## Comparing the Models

Although the models were similarly in their ability to predict values for `Stroke_Analysis_Test`, it appears that the `Big_model` is a better predictive model. This is because it exhibited a smaller mean absolute prediction error than the `Small_model` as well as an R squared value that was greater than that of the `Small_model`. These indicate that the `Big_model` was better able to predict the BMI value of a patient included in the `Stroke_Analysis_Test` data set and that the variation in BMI is better explained by this model. 

# Discussion

## The Chosen Model {.unnumbered}

It was decided that the `Big_model` should be the chosen model for this study.  This is because it was demonstrated to be better fitted to the training sample (`Stroke_Analysis_train`), and better able to predict values of BMI for the test sample (`Stroke_Analysis_Test`). It was found to have both a smaller AIC and a smaller BIC value in the in-sample comparison than the `Small_model`, indicating that it has a greater quality of fit. This was further demonstrated in residual vs. fitted plots. Furthermore, the `Big_model` was found to have a smaller MAPE value than the `Small_model` when it was used to predict the values of `Stroke_Analysis_Test`, showing that it had the ability to predict BMI with less error. Finally, the `Big_model` had a greater R squared value than the `Small_model`, demonstrating that it was able to explain a larger amount of the variation in bmi in the `Stroke_Analysis_Test` data. Importantly, the `Big_model` also did not violate any assumptions for linear regression, as it was seen in the residual plots that it demonstrated linearity, independence, equal variance, and normality in its distribution. Had it violated any of these key assumptions its utility as a linear model would be severely compromised. 

## Answering the research question {.unnumbered}

The research question for this analysis was:

**How effectively can the body mass index of a hospital patient who has been evaluated for a stroke be predicted by the average blood glucose level of the patient that was recorded in their EHR and how is the quality of prediction changed when adjusting for age, employment type, residence type, stroke history, and smoking status?**

To answer this question, both the `Small_model` and the `Big_model` need to be taken into consideration as the `Small_model` involves uses only the average blood glucose level as a predictor and the `Big_model` adjusts for age, employment type, residence type, stroke history, and smoking status. In the in-sample comparison, the `Big_model` was found to have been fit more closely to the values in the training data set than the `Small_model`, demonstrating that the model that was built with adjustment made for age, employment type, residence type, stroke history, and smoking status had smaller residual values (between predicted and observed BMI values) for the data it is built on than one that only uses the average blood glucose level as a predictor. This suggests that there is a possibility that BMI is able to be more effectively predicted when adjustments for age, employment type, residence type, stroke history, and smoking status are made, but further testing is required to make that statement confidently. For this reason, a model validation study was completed, in which the `Big_model` and the `Small_model` were compared in their ability to predict BMI values using data that they were not trained on. 

The model validation study applied each of the models to an unfamiliar data set of 986 new observations (named `Stroke_Analysis_Test`). Fitted and residual values for each model were calculated and compared using numeric summaries and plots. It was observed that the mean absolute prediction error for the model which adjusted for age, employment type, residence type, stroke history, and smoking status was lower than the MAPE for the model that only incorporated average blood glucose level as a predictor for BMI. Furthermore it was observed that the model in which age, employment type, residence type, stroke history, and smoking status were adjusted for had a higher R squared value than the model that only incorporated average blood glucose level as a predictor for BMI. Both of these observations further substantiated the previously mentioned possibility that the quality of prediction of BMI increased when average blood glucose level was used as a predictor *and* adjustments are made for age, employment type, residence type, stroke history, and smoking status. 

While it appeared that the quality of BMI prediction increased with the adjustments mentioned above, it is important to discuss the quality of prediction in general for both models. In the `Small_model`, where only average blood glucose was used as a predictor, it was found that average blood glucose demonstrated a very weak positive association with BMI, with an expected increase in the natural log of BMI of 0.00062 for every additional 1 mg/dl of blood glucose for in patient. *This means that the smaller model predicts that an extremely large increase or decrease in average blood glucose is required to create even a modest change in bmi*. Given this small demonstrated influence it cannot be stated confidently that average blood glucose as stated in a patient's EHR is an effective predictor of that patient's BMI.

In the `Big_model`, none of the coefficients for any of the variables used in the linear model had a value greater than 0.0147, indicating that a only a very large change (in many cases an unrealistically high change) in one of the quantitative variables would be expected to inspire a large change in the predicted BMI value of a patient. Importantly, a number of the coefficients for the variables used in this model had 90% confidence intervals that contained zero, indicating that the sample size of the data set used to create the model was not large enough to establish a positive or negative correlation between these variables and BMI with respect to an alpha level of .10.  None of the categorical variables were demonstrated to have the ability to change the predicted BMI value of a patient by a factor that would alter their status as underweight, healthy, overweight, or obese as per CDC guidance. For this reason, it can be said that although a marginally better quality of prediction is observed when predicting BMI using average blood glucose level *and* adjustments are made for age, employment type, residence type, stroke history, and smoking status, **the quality of prediction is still very low**. R squared values in the validation study demonstrated that both the `Big_model` and the `Small_model` were each only able to explain around 4% of the change in BMI within the `Stroke_Analysis_Test` data set. 

Given each of the points made above, it appears that the research question is best answered as follows:

**The body mass index of a hospital patient who has been evaluated for a stroke cannot be effectively predicted by the average blood glucose level that was recorded in their EHR. However, slightly more accurate predictions of BMI for hospital patients who have been evaluated for stroke can be calculated by using average blood glucose level of the patient as a predictor *and* adjusting for age, employment type, residence type, stroke history, and smoking status of the patient.**

My pre-analysis expectations were that the the body mass index of a hospital patient who has been evaluated for stroke status would be able to be effectively predicted by the average blood glucose level of that patient and that adjusting for age, employment type, residence type, stroke history, and smoking status would improve the predictive quality. Although I had correctly anticipated that the inclusion of additional variables would improve the predictive quality, the weak predictive ability of the average blood glucose level was not at all expected. 

There are a number of limitations to this study. First, the sample size used to train the model was limited, which in turn limited the ability of both of the models to be to reflect the true distribution of data for patients who have been evaluated for stroke. Second, the models created were strictly linear and thus it is possible that although average blood glucose cannot be used in a linear model to predict BMI adequately, it may be a useful predictor when incorporated into a different type of regression model (such as a parabolic or cubic model). 

## Next Steps {.unnumbered}

In looking to create a prediction model for BMI using electronic health record data, there are a number of steps that may be taken in the future to create a more comprehensive model with a higher quality of BMI prediction. One item to consider is the amount of observations used to train the model. Because in 2022 the use of electronic health record systems is fairly ubiquitous, it is not unrealistic to consider that data from far more than 2299 patients could be used to create a predictive model for BMI. Additionally, more variables may be considered in making this model, specifically those that are known to modulate the mass of a person, such as participation in exercise or physical disability. 

Additionally, one may consider using an approach to predicting BMI that does not involve a linear model, but rather using cubic or quadratic models to account for the possibility that different variables exert different effects on a patient's BMI based on where that person lies on the distribution of BMI values. 

Finally, further studies looking to establish a predictive model for BMI may consider looking beyond only patients who have been evaluated for a stroke as per their electronic health records. This may involve looking at populations that have been evaluated for other diseases or looking at a broader spectrum of hospital patients with less specific constraints. It is possible that people who have been evaluated for a stroke do not have particularly strong correlations between BMI and the variables examined in this study, but that other populations do. 
 
## Reflection {.unnumbered}

Reflecting on my approach to this study knowing what I do now, there are a few of things that had I known at the beginning of the study, I would have done differently. 

Before I completed this project, I was not aware of how I could display the confidence intervals for the coefficients of a linear model created using `ggplot2`. In completing this project I found it to be an incredibly useful way to visualize the influence of variables in a linear model and a useful tool for explaining an equation. If I had understood how to do this at the beginning of the project I would have created a plot of coefficients and confidence intervals before writing any explanation of my model so that I could have more quickly and effectively described its equation and anticipated the effects of each of its variables.  

Additionally, when I began working on this project, I did not have a strong understanding of the `car` package and how it can be used to analyze the the Variance Inflation Factor in table form  to assess collinearity of a multiple linear regression model. When I began the project, I was planning on only using pearson correlations and scatter plot matrices to analyze collinearity, however, if I had been better acquainted with the `vif` function, I may have chosen additional variables for my big model, as I would more closely be able to examine whether or not there was collinearity between any of my predictors. 


# Session Information {.unnumbered}

```{r}

sessioninfo::session_info()

```
